Finite Strain Behavior of Polyurea
for a Wide Range of Strain Rates
by
Jongmin Shim
M.S., Massachusetts Institute of Technology (2005)
M.S., Korea Advanced Institute of Science of Technology (2001)
B.S., Korea Advanced Institute of Science of Technology (1998)
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
ARCHIVE
Doctor of Philosophy in the field of Engineering Mechanics
at the
MASSACHUSETTS INSTrTUTE
OF TECHNOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MAR 2 3 2010
February 2010
LIBRARIES
@ 2009 Massachusetts Institute of Technology. All rights reserved.
Signature of Author..............
Department of Civil and Environmental Engineering
November 16, 2009
... ....
Tomasz Wierzbicki
epartment of Mechanical Engineering
.. .
...
. ................
Certified by
Professor of Applied Mechanics,
Thesis Supervisor
.....................
Dirk Mohr
se rch Professor, Ecole Polytechnique
Thesis Supervisor
A
Certified by .
CNRS Assistant
.
C ertified by .........................
. . .......... ..
.
.
. .. . . . . . . . . . . .
. . . . . . . ..
Eduardo Kausel
Profes or of Civil and Environmental En ineering
A 4Aesis Reader
Accepted by........
!
Daniele Veneziano
Chairman, Departmental Committee for Graduate Students
.................
Finite Strain Behavior of Polyurea
for a Wide Range of Strain Rates
by
Jongmin Shim
Submitted to the Department of Civil and Environmental Engineering
on November 16, 2009, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in the field of Engineering Mechanics
Abstract
Polyurea is a special type of elastomer that features fast setting time as well as good chemical
and fire resistance. It has also good mechanical properties such as its high toughness-to-density
ratio and high strain rate-sensitivity, so its application is recently extended to structural purpose
to form sandwich-type or multi-layered plates. Those structures can be used for retrofitting of
military vehicles and historic buildings, absorbing energy during structural crash.
In order to investigate its behavior of hysteresis as well as rate-sensitivity, three different
testing systems are used to cover a wide range of strain rates up to strain of 100%. In view
of impact and blast events, the virgin state of polyurea is considered throughout the experiments. First, a hydraulic universal testing machine is used to perform uniaxial compressive
loading/unloading tests in order to investigate its hysteresis behavior at low strain rates (0.001/s
to 10/s). Second, two distinct gas-gun split Hopkinson pressure bar [SHPB] systems are employed to cover high strain rates: a nylon bar system (700/s to 1200/s) and an aluminum bar
system (2300/s to 3700/s). Lastly, the rate-sensitivity for intermediate strain rates (10/s to
1000/s) is characterized using a modified SHPB system. The device is composed of a hydraulic
piston along with nylon input and output bars.
A finite strain constitutive model of polyurea is presented in order to predict the hysteresis
and rate-sensitivity behavior. The 1-D rheological concept of two Maxwell elements in parallel
is employed within the framework of the multiplicative decomposition of the deformation gradient. Model parameters are calibrated based on the uniaxial compressive tests at various rates.
The corresponding algorithms is implemented as a user-defined material subroutine VUMAT
for ABAQUS/Explicit, and used to predict the response of polyurea. The proposed constitutive model reasonably captures the experimentally observed asymmetric rate-sensitivity and
stress-relaxation behavior: strong rate-sensitivity and large amount of stress relaxation during loading phase, but weak rate-sensitivity and smaller amount of stress relaxation during
unloading phase. In order to validate the proposed model, various dynamic punching tests
are performed, and their results are well compared with the model predictions during loading
although the prediction of unloading behavior can be further improved.
Thesis Supervisor: Tomasz Wierzbicki
Title: Professor of Applied Mechanics, Department of Mechanical Engineering
Thesis Supervisor: Dirk Mohr
Title: CNRS Assistant Research Professor, Ecole Polytechnique
Thesis Reader: Eduardo Kausel
Title: Professor of Civil and Environmental Engineering
Acknowledgments
I would like to express my deep gratitude to Professor Tomasz Wierzbicki for taking me into the
Impact and Crashworthiness Lab and providing me with much needed support, guidance, and
mentorship. He has enlightened me on how important to keep an engineering mind. My sincere
thanks are due Professor Dirk Mohr at
Ecole Polytechnique for his mentorship and his valuable
insight into my research. He has shown me an excellent example for how to do research. I would
also like to offer many thanks to Professor Eduardo Kausel, Professor Jerome J. Connor, and
Professor David Roylance for their participation in my thesis committee and for their helpful
comments. Moreover, I am indebted to Professor Gerard Gary at
Ecole Polytechnique for his
guidance into the split Hopkinson pressure systems. The financial support of this work through
the Office of Naval Research is also greatly acknowledged.
I would like to thank all the members and alumni of the ICL (Dr. Young-Woong Lee, Dr.
Xiaoqing Teng, Dr. Liang Xue, Dr. Li Zheng, Dr. Yuanli Bai, Dr. Carey Walters, Dr. Yaning
Li, Ms. Allison Beese, Mr. Meng Luo, Mr. Matthieu Dunand, Ms. Danielle Issa, and Ms.
Kirki Kofiani) for creating a cooperative working environment. My thanks are also extended
to Ms. Sheila McNary for her administrative assistance.
Life at MIT would be much less interesting without several individuals: previous officemates (Georgios, Emilio, Matt/Cara and Chris), Korean colleagues in CEE (Joonsang-hyung,
Sungjune-hyung, Yunseung-nuna, Jungwuk-hyung, Phillseung-hyung, Sanghyun-hyung, Sangyoonhyung, Hongchul-hyung and many others), and colleagues from the neighboring labs (Yeunwoohyung, Heejin, Hyunjoe-hyung and Shawn). It was a privilege to know all of you.
Finally, my deepest gratitude goes to my family: my wife, Eunkyung, for her constant
support and encouragement; my son, Seohyun, for providing me with ineffable joy of life; and
my parents and sister for their unconditional love and faith in me. Thank you very much.
Contents
1
2
12
Introduction
. . . . . . . . . .
. . . . . . . . . . . . . . 12
. . . . . . . .
1.1
M otivation
1.2
Objective and Tasks...... . . . . . . . . . . . .
1.3
Outline of Dissertation..... . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 14
. . . . . 14
17
Experimental Work
Introduction
2.2
Experimental Procedures........
2.3
2.4
2.5
. . . . ..
17
. . . . . . . . . . . . . . . . . . . . ...
19
. .. . .. . . .. . ..
....................
2.1
2.2.1
Universal Testing M achine . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2
Conventional SHPB Systems......
2.2.3
Modified SHPB with Hydraulic Actuator
2.2.4
Determination of the Stress-Strain Curves.. .
Experimental Results......... . . . . .
.. . .. . ..
........
. . . 20
. . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . ..
26
. . . . . . . . . . 27
. . . . . . . . . .
. . . . . . . . . . . . . 27
2.3.1
Experiments using the Universal Testing Machine
2.3.2
Experiments Using Conventional SHPB Systems . . . . . . . . . . . . . . 29
2.3.3
Experiments Using the Modified SHPB Systems
2.3.4
Comment on the Signal Oscillations....... . . . . . .
. . . . . . . . . . . . . . 33
Discussion...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 34
. . . 37
. . . . . . . . . . . . . . . 37
2.4.1
Experimental Results.... . . . . . . . . . .
2.4.2
Intermediate Strain Rate Testing Systems . . . . . . . . . . . . . . . . . . 38
C onclusion
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 41
3
Constitutive Modeling
Introduction................
3.2
Experimental Investigation...........
3.4
3.5
3.6
. . . . . . . 42
. . .
. .. . .. . . .. . ..
3.1
3.3
4
42
. . . . . . . . 45
.. . .. . . . . . .
. . . . . . . . . . . 45
. .. . . . . . . . . . .
3.2.1
Material.............
3.2.2
Relaxation Experiments.......... . . . . . . . . . .
3.2.3
Continuous Compression Experiments... . . .
3.2.4
Stair Compression Experiments....... .
. . . . . . . . 45
. . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . ..
. . . . . . . 49
.. . . .. . . . . . .
Constitutive Model................
. . . . . .
. .. . .. . .. . . .
..............
. . . 49
3.3.1
Motivation
3.3.2
Homogenization............ . . . . . . . . . . . . . . .
3.3.3
Constitutive Equations for Volumetric Deformation . . . . . . . . . . . . . 53
3.3.4
Maxwell Model for Isochoric Deformation . . . . . . . . . . . . . . . . . . 54
3.3.5
Specialization of the Maxwell Model for Network A . . . . . . . . . . . . 56
3.3.6
Specialization of the Constitutive Equations for Network B . . . . . . . . 57
. . . . . 52
Identification of the Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1
Constitutive Equations for Uniaxial Loading..
3.4.2
Model Calibration........... . . . . . . . .
Comparison of Simulation and Experiments.. . . .
. . . . . . . . . . . . ..
59
. . . . . . ..
61
. . . .
. . . . . . . . . . . . . . . 65
. . . . . . . . . . . . . . . . . . . 65
3.5.1
Continuous Compression... . . . . .
3.5.2
Stair Compression...... . . .
3.5.3
Relaxation...... . . . . . . . .
3.5.4
Discussion............ . . . . . . . . . . . . . . . . . . . . .
Conclusions.
.
. . . . . . . . . . . . . . . . . . . . ..
... .....................
......
4.1
Introduction..............
4.2
Punch Experiments...............
67
. . . . . . . . . . . . . . . . . . . . . 70
. . . 70
. ..
71
73
Validation Application
4.3
47
. . ..
. . . . . . . .
. . .. . . . .
. . . . . . . . . . . . . 73
. . . . . . . . . . ..
75
. . . . . . . . . . . . . . 75
4.2.1
Specimens......... . . . . . . . . . . . . .
4.2.2
Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.3
Experimental Results...........
. .. . .. . ..
Constitutive Model...... . . . . . . . . . . . . . . . . . .
. . . . . . . ..
76
. . . . . . . . . . . 79
5
4.3.1
Response of N etwork A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.2
Response of N etwork B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.3
Model Parameter Identification............
. . .. .
. . . . . ..
83
4.4
Numerical Simulations of the Punch Experiments . . . . . . . . . . . . . . . . . . 85
4.5
Discussion on Unloading Behavior of Polyurea.... . . . . . . . . . . . .
. . . 87
4.6
Conclusion.............
.. . .. . . .. . . . . . . . . . . . . . .
. . . 93
95
Conclusion and Suggestions
5.1
Summary of M ain Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2
Suggestions for Future Studies........ . . . . . . . .
. . . . . .
. . . . . . 96
A Identification of the Wave Propagation Coefficient for Viscoelastic Bars
B List of Papers with Reference to Respective Chapters
98
100
List of Figures
. . . . . . . . . . . . . ...
2-1
(a) Conventional and (b) Modified SHPB systems.
2-2
Identification of Poisson's ratio from the linear relationship between the logarith-
21
mic radial and axial strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2-3
Test results from the universal testing machine: (a) True stress-strain curves, (b)
True strain versus true strain curves . . . . . . . . . . . . . . . . . . . . . . . . . 29
2-4
Propagation coefficient of aluminum bar (first row) and nylon bar (second row).
(a)-(c) Longitudinal wave speed and (b)-(d) Attenuation coefficient...
2-5
. ...
30
Comparison of forces between input bar and output bar from SHPB tests from
aluminum bar tests: (a) 3700/s, (b) 2300/s; and nylon bar tests: (c) 1200/s, (d)
700/s
...........
. . . . . . . . . . . . . . . . . . ..
. . . . . . . . . . . 32
. . . 33
2-6
(a) True stress-strain curves, (b) True strain rate versus true strain curves.
2-7
Comparison of forces between input bar and output bar from the modified SHPB
tests: (a) 1000/s, (b) 110/s, (c) 36/s and (d) 10/s . . . . . . . . . . . . . . . . . 35
2-8
Test results from the modified SHPB system: (a) True stress-strain curves, (b)
True strain rate versus true strain curves.
2-9
. . . . . . . . . . . . . . . . . . . . . . 36
Comparison of the results from the modified SHPB with those from other two
testing methods: (a) True stress-strain curves, (b) True strain rate versus true
strain curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2-10 True stress as a function of the strain rate at selected strain levels: (a) Results
of the current study and a fit of Eq. (2.26) to the results from the present study.
(b) Comparison of the results with previous studies.
. . . . . . . . . . . . . . . . 38
3-1
Results of relaxation tests (left column) and corresponding simulations (right
column). (a)-(b) True stress histories, (c)-(d) Relaxation moduli histories, (e)(f) Isochronous stress-strain curves for different instants after the rapid strain
loading. .......
3-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
. .. .
Results of continuous loading/unloading experiments and corresponding simulations. (a) History of applied strain. Note that time in the x-axis is normalized by
the total testing duration
leo/eoI
where
ao
-
10-
, 10-2, 10-1, 100, and 101 /s
(b) Stress-strain curves obtained from experiments and
3-3
.
. . . . . . . . . . . . . . 48
Results for stair compression loading/unloading. (a) History of applied strain.
Note that time in the x-axis is normalized by the total testing duration
where Eo = 1 and
o = 10-3
, 10-2,
Ieo/eol
1
10-1, 100, and 10 /s . (b) Stress-strain
curves from experiments and (c) Corresponding simulations. (d) Comparison of
results from continuous and stair-type loading/unloading experiments and (e)
Corresponding simulations......... . . . .
3-4
. . . . . . . . . . . . .
Estimation of the behavior of Network A using the equilibrium path concept for
path concept for different strain rates: (a) 10
100 /s and (e) 10 1 /s........
3-5
. .. . . . . .
3
/s, (b) 10- 2 /s, (c) 10- 1/s, (d)
. . . . . . . . . . . . . . . . . . . 51
Proposed rheological model for polyurea composed of two Maxwell elements in
parallel................
3-6
. . . . . 50
. .. . .. . . . . . . . . . . . . .
. . . . . . . 52
Material model parameter identification for Network A. (a) Estimated experimental stress-strain curve for Network A; (b) Equivalent Mandel stress as a
function of the scalar deformation measure (; (c) HA as function of the equivalent viscous deformation rate dA for the identification of the model parameters
PA and nA through power-law fit.
3-7
. . . . . . . . . . . . . . . . . . . . . . . . . . 63
Material model parameter identification for Network B.
(a) Estimated exper-
imental stress-strain curve for Network B; (b) Equivalent Mandel stress as a
function of the scalar deformation measure (; (c) (B as function of the equivalent viscous deformation rate dB; identification of the model parameters QB and
n..............................
. . . . . . ...
66
3-8
Comparison of simulation results and experiments for continuous loading-unloading
cycles. (a) 10- 3 /s, (b) 10-
3-9
/s, (c) 10-1/s, (d) 100 /s and (e) 101/s
. . . . . . . . 68
Comparison of simulation results and experiments for stain loading/unloading.
(a) 10
4-1
2
3
/s, (b) 10- 2 /s, (c) 10-1/s, (d) 100 /s and (e) 10s
.
. . . . . . . ...
69
Photos of the experiments with (a) the small punch and (b) the large punch.
The second row shows the set up for (c) free boundaries in all lateral directions
and (d) for constrained boundary conditions in the width direction. . . . . . . . . 77
4-2
Applied loading profiles: (a) Applied velocity history; the velocity axis is normalized by the applied initial velocity of either vo
1mm/s or 100m/s; the
time axis is normalized by |2uo/vol with uo = 7mm; (b) Corresponding applied
displacement history........... . . . . . . . . . . . . . . . . .
4-3
. . . 77
. . . .
Measured load-displacement curves for experiments with (a) the small punch,
(b) the large punch.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
. . . . 79
4-4
Rheological model of the rate dependent constitutive model for polyurea.
4-5
Comparison of simulation results and experiments for continuous loading-unloading
cycles. (a) 10-3/s, (b) 10- 2/s, (c) 10- 1/s, (d) 100 /s, (e) 10 1 /s. . . . . . . . . . . 84
4-6
The contour plots of the logarithmic strain in thickness-direction from simulations with vo = 100m/s at an indentation depth of 7mm: (a) small hemispherical
indenter, and (b) large hemispherical indenter. The detail in Figure 4-6a shows
the locations for which the strain rates dA and dB are plotted in Figure 4-8.
4-7
. . 86
Comparison of simulations and experiments. Force-displacement curves for (a)
small punch with free lateral boundaries, (b) large punch with the free lateral
boundaries, (c) large punch with constraint in the width direction.
4-8
. . . . . . . . 88
Results from the small punch simulation with vo = 100mm/s and free lateral
boundary conditions at three different locations (labeled by A, B and C in
Figure 4-6(a): Histories of (a) the strain-like deformation measure, (, (b) the
viscous strain rate of Network A, dA, (c) the viscous strain rate of Network B,
dB. Note that the loading direction is reversed at t = 0.14s.
4-9
. . . . . . . . . . . 89
Stress-strain curves for a single loading-unloading cycle at e = 10- 2/s as obtained
from (a) experiments and (b) simulations.
. . . . . . . . . . . . . . . . . . . . . 90
4-10 Comparison of simulation results and experiments for continuous loading/unloading
cycles at
a=
10-2/s. (a) ef = -0.5,
(b) ef = -1.0,
(c) ef = -1.5.
. . . . . . . . 92
4-11 Illustration of the Mullins effect. Five compression loading and unlading cycles
are performed at the constant strain rate of 10- 2 /s up to the maximum strain
of -1.0.
The stress is zero between subsequent cycles for about five minutes.
. . 93
Chapter 1
Introduction
1.1
Motivation
Polyurea has been generally used for coating purpose due to its good chemical properties such as
fast setting time (few minutes or less) and good water/chemical/fire resistance. Polyurea is used
on metallic substrates where it provides corrosion and abrasion resistance in harsh environments.
Moreover, it has also good mechanical properties such as high toughness for its low density and
high rate-sensitivity.
Thus, its application is recently extended to structural purpose to form
sandwich-type or multi-layered plates, which can be used for retrofitting of military vehicles,
historic buildings, gas/oil pipelines and marine structures, absorbing energy during structural
crash and holding metal/brick fragments even after structural failure. Like other elastomers,
polyurea is viscous material so its mechanical properties are rate dependent. Especially under
extreme loading conditions such as blast, projectile and explosive loadings, polyurea becomes
very attractive; its lightness is good for operational purpose, and its mechanical strength is
enhanced under extreme loading conditions. This structural purpose of polyurea under extreme
loading conditions motivates our research on a wide range of strain rates up to large strains.
In view of the simulation of blast and impact events, we limit our attention to the mechanical
behavior of polyurea in its virgin state.
Limited experimental study of virgin polyurea has been published on the rate-sensitivity
behavior for a wide range of strain rates.
Polyurea shows a highly nonlinear viscoelastic
behavior at finite strains (e.g. Amirkhizi et al., 2006, Bogoslovov and Roland, 2007, Roland
et al., 2007).
The stress-strain response of most polymeric materials shows a pronounced
strain rate sensitivity at low, intermediate and high strain rates. Various authors published
experimental results on the strain rate sensitive response of amorphous glassy polymers (e.g.
Chou et al., 1973, Boyce et al., 1988, Walley et al., 1989, Cady et al., 2003, Siviour et al., 2005,
Mulliken and Boyce, 2006, Mulliken et al., 2006), crystalline glassy polymers (e.g. Chou et al.,
1973, Bordonaro and Krempl, 1992, Cady et al.,2003, Siviour et al., 2005, Khan and Farrokh,
2006) and elastomers (e.g. Gray et al., 1997, Rao et al.,1997, Song and Chen, 2003, 2004, Hoo
Fatt and Bekar, 2004, Shergold et al., 2006, Roland, 2006) including polyurea (e.g. Amirkhizi
et al., 2006, Roland et al., 2007, Sarva et al., 2006). However, only few experimental studies
deal with the intermediate strain rate behavior of elastomers at large deformations. Sarva
et al. (2006) performed intermediate strain rate compression tests on polyurea for maximum
strains greater than 100% but the strain rates were only 14 ~ 80/s. The same research group
also obtained test results for a strain rate of 800/s using a very long aluminum SHPB system.
Roland et al. (2007) characterized the tensile behavior of polyurea over a strain rate range of
14 ~ 573/s and up to strains of more than 300%.
As for the experimental study on the rate-sensitivity of polyurea, little research has been
reported on modeling of virgin polyurea under loading/unloading conditions for a wide range
of strain rates. Finite viscoelasticity models of elastomers may be formulated using the socalled hereditary integral approach (Coleman and Noll, 1961, Bernstein et al., 1963, Lianis,
1963, McGuirt and Lianis, 1970, Leonov, 1976, Johnson et al., 1994, Haupt and Lion, 2002,
Amirkhizi et al., 2006) but their validity is often limited to a narrow range of strain rates
(Yang et al., 2000, Shim et al., 2004, Hoo Fatt and Ouyang, 2007). As an alternative to the
hereditary integral approach, the framework of multiplicative decomposition of the deformation
gradient (Kroner, 1960 and Lee, 1969) is frequently used in finite viscoelasticity (e.g. Sidoroff,
1974, Lubliner, 1985, Le Tallec et al., 1993, Reese and Govindjee, 1998, Huber and Tsakmakis,
2000). In that framework, the nonlinear viscoelasticity of elastomers is commonly described
through a rheological spring-dashpot models of the Zener type (e.g. Roland, 1989, Johnson
et al., 1995, Bergstrom and Boyce, 1998, Huber and Tsakmakis, 2000, Quintavalla and Johnson, 2004, Bergstrom and Hilbert, 2005, Qi and Boyce, 2005, Areias and Matous, 2008, Hoo
Fatt and Ouyang, 2008, Tomita et al., 2008). As for the hereditary integral approach based
models, however, most multiplicative decomposition based models have been also experimentally validated for a narrow range of strain rates. Quintavalla and Johnson (2004) adopted the
Bergstrom-Boyce model to describe the dynamic behavior of cis-(1,4) polybutadiene at high
strain-rate of 3000/s to 5000/s. Recently, Hoo Fatt and Ouyang (2008) proposed a thermodynamically consistent constitutive model with modified neo-Hookean rubber elastic springs
to describe the steep initial stiffness of virgin butadiene rubber under tensile and compressive
loading at intermediate strain rates (76/s to 450/s).
1.2
Objective and Tasks
The objective of the present dissertation is to develop a finite constitutive model of virgin
polyurea for a wide range of strain rates under loading/unloading condition.
The material is
assumed to be isotropic, and the isothermal condition is considered at the room temperature.
In order to achieve the objective, the following tasks are set:
* To characterize the mechanical properties of virgin polyurea at low, intermediate and high
strain rates for large strains;
" To develop a finite constitutive model of virgin polyurea;
" To demonstrate the validity of the proposed constitutive model by performing dynamic
punching tests and simulations.
1.3
Outline of Dissertation
This dissertation is composed of five chapters.
Excluding Chapter 1 and Chapter 5, each
chapter addresses one specific topic and it is self-contained because it has already been published
or submitted for publications. A list of publications related to this dissertation is presented in
Appendix B.
Chapter 2 presents experimental results on polyurea. The strain rate sensitivity of polyurea
is characterized using a modified split Hopkinson pressure bar (SHPB) system. The device is
composed of a hydraulic piston along with nylon input and output bars. In combination with
an advanced wave deconvolution method, the modified SHPB system provides an unlimited
measurement time, and thus can be used to perform experiments at low, intermediate and high
strain rates. A series of compression tests of polyurea is performed using the modified SHPB
system. In addition, conventional SHPB systems as well as a universal hydraulic testing machine
are employed to confirm the validity of the modified SHPB technique at low and high strain
rates. The analysis of the data at intermediate strain rates shows that the strain rate is not
constant due to multiple wave reflections within the input and output bars. It is demonstrated
that intermediate strain rate SHPB experiments require either very long bars (> 20m) or very
short bars (< 0.5m ) in order to achieve an approximately constant strain rate throughout the
entire experiment.
Chapter 3 is devoted to a constitutive model for polyurea based on the experimental results.
Continuous loading and unloading experiments are performed at different strain rates to characterize the large deformation behavior of polyurea under compressive loading. In addition,
uniaxial compression tests are carried out with stair-like strain history profiles. The analysis
of the experimental data shows that the concept of equilibrium path may not be applied to
polyurea. This finding implies that viscoelastic constitutive models of the Zener type are no
suitable for the modeling of the rate dependent behavior of polyurea. A new constitutive model
is developed based on a rheological model composed of two Maxwell elements. The soft rubbery response is represented by a Gent spring while nonlinear viscous evolution equations are
proposed to describe the time-dependent material response. The eight material model parameters are identified for polyurea and used to predict the experimentally-measured stress-strain
curves for various loading and unloading histories. The model provides a good prediction of
the response under monotonic loading over wide range of strain rates, while it overestimates
the stiffness during unloading. Furthermore, the model predictions of the material relaxation
and viscous dissipation during a loading/unloading cycle agree well with the experiments.
Chapter 4 presents the validation application for the proposed constitutive model. Punch
indentation experiments are performed on 10mm thick polyurea layers on a steel substrate.
A total of six different combinations of punch velocity, punch size and the lateral constraint
conditions are considered. Furthermore, the time integration scheme for a newly-developed ratedependent constitutive material model is presented and used to predict the force-displacement
response for all experimental loading conditions. The comparison of the simulations and the
experimental results reveals that the model is capable to predict the loading behavior with good
accuracy for all experiments which is seen as a partial validation of the model assumptions
regarding the pressure and rate sensitivity. As far as the unloading behavior is concerned, the
model predicts the characteristic stiff and soft phases of unloading. However, the comparison
of simulations and experiments also indicates that the overall model response is too stiff. The
results from cyclic compression experiments suggest that the pronounced Mullins effect needs
to be taken into account in future models for polyurea to improve the quantitative predictions
during unloading.
Chapter 5 summarizes the main contributions of the dissertation, and presents suggestions
for future studies.
Chapter 2
Experimental Work
2.1
Introduction
Polyurea is a special type of elastomer which is widely used as coating material. It features a
fast setting time (few minutes or less) as well as good chemical and fire resistance. Polyurea
is frequently used on metallic substrates where it provides corrosion and abrasion resistance
in harsh environments. Applications include transportation vehicles, pipelines, steel buildings
or marine constructions. More recently, polyurea is also considered for the blast protection
of transportation vehicles because of its high toughness-to-density ratio, in particular at high
strain rates. It is the objective of this work to characterize the mechanical properties of polyurea
at low, intermediate and high strain rates.
The mechanical properties of most metallic engineering materials exhibit only a weak ratedependence at strain rates below 100/s. Therefore, metals are usually tested either at very
low strain rates (< 10
2
/s) on universal testing machines or at high strain rates (> 102 /s) on
split Hopkinson pressure bar (SHPB) systems. The stress-strain response of most polymeric
materials on the other hand shows a pronounced strain rate sensitivity at low, intermediate
and high strain rates. At small strains, the viscoelastic properties of polymers are typically
determined using dynamic mechanical analysis (e.g. McGrum et al. 1997). The characterization
of the large deformation response of polymers at low and intermediate strain rates of up to 10/s
can be performed on hydraulic testing systems (e.g. Yi et al. 2006, Song et al. 2007). As for
metals, conventional SHPB systems are employed to characterize the large deformation response
of polymeric materials at high strain rates. However, as discussed by Gray and Blumenthal
(2000), low impedance Hopkinson bars are recommended when testing soft polymeric materials
(e.g. Zhao et al., 1997, Chen et al., 1999, Sharma et al., 2002). Hoo Fatt and Bekar (2004)
developed a pulley system to perform large strain tensile tests on rubber sheets at intermediate
and high strain rates. Inspired by this work, Roland et al. (2007) designed a pendulum impact
tester to study the tensile properties of elastomers at strain rates of up to about 500/s. In
both testing systems, the issues related to the strain measurements under dynamic loading
conditions are circumvented through the use of digital image correlation (DIC) based on high
speed camera recordings.
Unlike for high strain rate experiments, the duration of the experiment poses a major
challenge when using SHPB systems for intermediate strain rate testing.
The experiment
duration Texp is given by the ratio of the strain Emax at the end of the experiment and the
average strain rate
e,
Texp = Emax/s. In order to avoid the superposition of waves, the maximum
duration of reliable measurements is limited to the input bar transit time. The input bar
transit time is an intrinsic property of the input bar and can only be lengthened by increasing
the bar length or by choosing a bar material of low wave propagation speed. In combination
with two strain measurements on each Hopkinson bar, wave separation techniques may be
used to overcome this limitation for elastic (e.g. Lundberg and Henchoz, 1977, Yanagihara,
1978, Park and Zhou, 1999) and viscoelastic bar systems (e.g. Zhao and Gary, 1997, Bacon
1999, Casem et al., 2003). However, Jacquelin and Hamelin (2001, 2003) as well as Bussac et
al. (2002) have shown that so-called two-point measurement wave separation techniques are
sensitive to noise. This finding led to the development of a mathematical framework for an
advanced wave deconvolution technique which is based on redundant measurements (Bussac et
al., 2002). Othman and Gary (2007) demonstrated the applicability of this testing technique to
the intermediate strain rate testing of aluminum on a hydraulic actuator driven SHPB system.
Othman et al. (2009) also employed this technique when using a 0.82m long bar to measure
the axial forces in a modified servo-hydraulic machine. In the present work, we make use of a
similar testing system as Othman and Gary (2007) to characterize the intermediate strain rate
response of the elastomeric material polyurea under compressive loading.
Various authors published experimental results on the strain rate sensitive response of amor-
phous glassy polymers (e.g. Chou et al., 1973, Boyce et al., 1988, Walley et al., 1989, Cady
et al., 2003, Siviour et al., 2005, Mulliken and Boyce, 2006, Mulliken et al., 2006), crystalline
glassy polymers (e.g. Chou et al., 1973, Bordonaro and Krempl, 1992, Cady et al., 2003, Siviour
et al., 2005, Khan and Farrokh, 2006) and elastomers (e.g. Gray et al., 1997, Rao et al., 1997,
Song and Chen, 2003, 2004, Hoo Fatt and Bekar, 2004, Shergold et al., 2006, Roland, 2006) including polyurea (e.g. Amirkhizi et al., 2006, Roland et al., 2007, Sarva et al., 2007). However,
only few experimental studies deal with the intermediate strain rate behavior of elastomers at
large deformations. Sarva et al. (2007) performed intermediate strain rate compression tests
on polyurea for maximum strains greater than 1.0, but the strain rates were only 14 ~ 80/s.
The same research group also obtained test results for a strain rate of 800/s using a very long
aluminum SHPB system. Roland et al. (2007) characterized the tensile behavior of polyurea
over a strain rate range of 14 ~ 573/s and up to strains of more than 3.0. In the present study,
an attempt is made to cover a similar range of strain rates by using the modified SHPB system
of Zhao and Gary (1997) in combination with the deconvolution method of Bussac et al. (2002)
to perform compression experiments on polyurea.
This paper is organized as follows. Section 2.2 describes all experimental procedures, notably
the conventional SHPB and the modified SHPB systems. The experimental results on polyurea
are presented in Section 2.3, followed by a discussion of the limitations of the present testing
system in Section 2.4.
2.2
Experimental Procedures
Three different testing systems are used to cover a wide range of strain rates: a universal testing
machine, a conventional SHPB system, and a modified SHPB system with a hydraulic actuator.
Throughout our presentation of the experimental methods, we use the hat symbol to denote
the Fourier transforms
2.2.1
f (w) =
f_
f
(t) e-tdt of time-dependent functions
f (t).
Universal Testing Machine
A hydraulic universal testing machine (Model 8800, Instron, Canton MA) is used to perform
compression tests at low and intermediate strain rates (10-2 ~ 10/s). The position of the
vertical actuator is controlled using the software MAX (Instron, Canton). The axial force F (t)
is measured using a low profile load cell of a maximum loading capacity of 10kN (MTS, Chicago,
IL) that has been positioned at a distance of 25mm from the specimen. At the same time, the
cross-head displacement is measured using an LVDT positioned at a distance of about 1300mm
above the specimen (integrated in the actuator piston).
A DIC system (Vic2D, Correlated
Solutions, Columbia, SC) is employed to measure the displacements ui2 , (t) and uout (t) of the
top and bottom loading platens, respectively. Furthermore, we make use of the DIC system to
quantify the Poisson's ratio of polyurea. Both a thin polymer layer (Teflon) and grease are used
to minimize the frictional forces at the contact surface between the specimen and the loading
platens.
2.2.2
Conventional SHPB Systems
Two distinct conventional SHPB systems are used in this study:
" The first is an aluminum bar system with a 1203mm long striker bar. Experiments of a
maximum duration of Texp = 472ps can be performed on this system.
" The second SHPB system is composed of thermoplastic nylon bars with a 1092mm long
striker bar. Thus, a maximum duration of Tep = 12 5 5 ts is achieved on that system.
Technical details of these systems are given in Table 2.1.
The gages for strain history
recordings are positioned near the center of the input bar and near the output bar/specimen
interface (Table 2.1, Figure 2-la). Using viscoelastic wave propagation theory (e.g. Zhao and
Gary, 1997), we reconstruct the incident and reflected waves based on the input bar strain
gage recordings to estimate the force Fi2 (t) and displacement uin (t) at the input bar/specimen
interface. Analogously, the force Fst (t) and displacement unut (t) at the output bar/specimen
interfaces are calculated after reconstructing the transmitted wave based on the output bar
strain gage recording.
2.2.3
Modified SHPB with Hydraulic Actuator
The total duration of the loading pulse in an experiment on a conventional SHPB system is
limited by the length of the striker bar. Thus, it is usually impossible to reach large strains at
strain
gage
strain
gage
II
ve
input bar
striker
II
T
specimen
output bar
(a)
output bar
fixed end
support
displacement
sensor
sensor
(b)
Figure 2-1: (a) Conventional and (b) Modified SHPB systems.
Table 2.1: Specifications of the conventional SHPB sytems
Nylon Bar System
Aluminum Bar System
Striker Input bar Output bar Striker Input bar Output bar
Length, L [m]
1.203
2.991
1.850
1.092
3.070
1.919
Radius, R [m]
20
20
20
16.5
20
20
Longitudinal wave
speed, co (w = 0)
5100
5100
5100
1740
1740
1740
2820
2820
2820
1187
1162
1145
-
1.493
0.335
-
1.537
0.394
[m/s)
Mass density, p [kg/m 3]
Distance between strain
gauge and specimen/bar
interface, d [m]
intermediate strain rates on conventional SHPB systems. To overcome this key limitation, we
make use of the modified SHPB system proposed by Zhao and Gary (1997). By substituting
the striker bar through a hydraulic actuator, almost infinite loading pulse durations may be
achieved. Having this setting, the right end of the output bar needs to be fixed in space as
its inertia is no longer sufficient to support the specimen (Figure 2-1b). In order to prevent
the failure of the nylon bars under excessive loads (it can be difficult to stop the piston), a
fixed end support system is designed such that the bars are released before elastic buckling
occurs.
Note that the wave superposition in the input and output bars can no longer be
avoided when the test duration Tezp exceeds the transit time for waves traveling from one
bar end to the other. Therefore, a wave separation technique is employed to reconstruct the
rightward and leftward travelling waves in the bars based on strain gage measurements. Once
both the rightward and leftward traveling waves in the bars are known, the interface forces and
velocities may be calculated using the same equations as those for the input bar in a conventional
SHPB system. Wave separation techniques in the time domain are efficient for non-dispersive
bars (e.g.
Lundberg and Henchoz, 1977), but these require more intense computations for
waves in dispersive systems (e.g. Bacon, 1999). Here, we adopt the frequency domain based
deconvolution technique of Bussac et al. (2002). In particular, displacement measurements are
included in addition to strain gage recordings.
Suppose that a strain wave e (x, t) in a bar is composed of the rightward traveling wave
ER (x, t) and the leftward traveling wave EL (x, t). In terms of Fourier transforms, we have the
multiplicative decomposition of the frequency and spatial dependence,
(X, W)
where
sR (xo, w) and
= ER
(Xo, W)
+ sL (TO, W) ei((w)(x-xo)
e(")(xxo)
(2.1)
L (xo, w) denote the Fourier transform of the respective strain histories
at some reference location xo. Moreover,
()
is the frequency-dependent wave propagation
coefficient for the respective bar system,
()
=
) + i (P)
with the frequency dependent wave number
c (W)
+ i (p)
(2.2)
n (w), the longitudinal wave propagation speed
c (w) = w/r (w), and the attenuation coefficient d (w).
The simplest approach to determine the functions
strain histories
sR (xo, w)
and sL (XO, W) is to measure the
E(xi, t) and e (X2, t) associated with the wave e (x, t) at two distinct locations
xi and x2 (on the same bar). Subsequently, one can solve the linear system of equations
b =Ax
(2.3)
with the unknowns
x
s
(W) =
(2.4)
W
[L (Xo, w)
= X::W
b(w)
the measurements
S(Xi, W)
(2.5)
and the coefficient matrix
A(w)=
eui
e-
1
;::WX
Go)(Xi-xo)
it(U)
ei(W)
(X2 -o)
(2.6)
(X2 -Xo)
However, the coefficient matrix becomes singular (det A = 0) if
(w) =
M
(2.7)
X2
-
X1
Bussac et al. (2002) propose an integration method in the complex domain to address this
problem. However, the same authors have also shown that the noise in the recorded strain gage
signals may still lead to erroneous solutions for x (w). In order to improve the solution of Eq.
(2.3) under the presence of measurement noise, it is useful to introduce redundant measurements
including force, velocity or displacement measurements. From Eq. (2.1), the Fourier transform
of force, velocity and displacement can be expressed as
F (x, w)
= E* (w) A [sR (zo, w) ei (w)(xxo) + eL (XO, W) i (w)(xxo)]
it (X, w) = c* (w) [R
(XO, W) e
(xo, w) ei((w)(x-o
±L
-(-
(2.8)
9)
t (x, w)
=
1* (w) [R (Xo, w) e-i())(xxO) -
sL (XO, U) i (w)(xxo)
(2.10)
and l* (w)
(2.11)
where
E* (w) = P
) =
,c*
((w))
=
()
( (P
In the present work, we perform only strain and displacement measurements.
Formally, we
write
(2.12)
b (QW)
f, (xQ~l,wC)
n (xQ+R, W)
where the subscripts
Q
and R represent the number of measurements for strains and displace-
ments, respectively, at the locations xi (i
=
1, ... ,
Q+
R) on the bar. The corresponding matrix
A (w) reads
A (w) =(.3
e-M (W)(*1-Xo)
ei(w)(X1-xo)
ed(VW)(XQ-Xo)
e i"()("4-_
l* (w)
e-i(W)(xQ+1-o)
l* (W) e~i5()(XQ+R -X0)
)
-*
(w) ei ()(XQ+1XO)
-1*
()
(2.13)
ei (*)(XQ+n-Xo)
For redundant measurements, the equation b = Ax for the unknown x is over-determined,
and cannot be solved exactly. Instead, an approximate solution is calculated by using the least
squares method to minimize the scalar error, e = jib - Ax|12
=
(b - Ax)H (b - Ax). Thus,
the approximate solution x minimizing the error must satisfy the equation AH
-
AHAX. As
long as the columns of A are linearly independent, the matrix AHA is positive definite (e.g.
Strang, 1985) and the unknown x can be determined as
x = [AHA)
lAHb
(2.14)
where the Hermitian AH (complex conjugate and transpose of A) corresponds to the transpose
of A if A is real (e.g. Magnus and Neudecker, 1988). Note that a least squares solution of
similar form has been presented by Hillstram et al. (2000) in the context of complex modulus
identification based on redundant strain measurements.
In order to rule out the linear dependence of the columns of A, we modify the propagation
coefficient
( (w)
artificially. In other words, when calculating A,
(
(w) is substituted by the
modified propagation coefficient ((w)
(W) =
(2.15)
)+ i
where / is a very small, but otherwise arbitrary, negative number; throughout our analysis,
we used
-10-7.
The modified propagation coefficient
perturbation of the propagation coefficient
(w) corresponds to the linear
(w) in the complex frequency domain. As a result,
the propagation coefficient is always complex-valued, and thus the singularity condition in Eq.
(2.7) can no longer be satisfied. Note that i/ introduces a very small artificial attenuation
(P) =
I/ + i d (P) + r/
(W) -
(2.16)
As a result, even purely elastic materials (a (w) = 0) exhibit some artificial attenuation (i.e.
non-zero imaginary value) which ensures the causality of the waves propagating in a bar (e.g.
Bacon, 1999).
In the present study we make use of a modified SHPB system with nylon input and output
bars. Table 2.2 summarizes the technical specifications of the testing system. Each bar is
equipped with three strain gages and a high contrast grid for optical displacement measurements
(Model 100H, Zimmer, Germany). After using Eq. (2.14) to determine the leftward and the
rightward traveling waves einL*(X
tan and
t)
e in / xin', t),) the displacement Uin (t) and the force
Fin (t) at the input bar/specimen interface are:
nin (w) = 1*(w) I (Xz,w)ei0(w)"N
Fin (w)
E* (w) A IR (x4", w) eif(w)x'
- L (XzI, w) e-i()n
(2.17)
+ " (zO", w) e-d(w)
(2.18)
Table 2.2: Specifications of the modified nylon SHPB sytem
Input bar Output bar
Length, L [im]
3.123
3.045
Radius, R [m]
20
20
Longitudinal wave
speed, co (w = 0)
1740
1740
1150
1150
0.61
0.825
1.515
1.523
2.623
2.582
0.953
2.183
[m/s]
Mass density, p [kg/m
3
]
Distance between strain
gauges and specimen/bar
interface, d5 g [in]
Distance between displacement
sensor and specimen/bar
interface, ddm [M]
Analogously, the displacement
neout (w)
and the corresponding force Fout (w) at the output
bar/specimen interface are determined from the output bar measurements.
2.2.4
Determination of the Stress-Strain Curves
The time histories of the displacements and forces at the specimen boundaries, ui, (t), uost (t),
Fin (t) and Fout (t) are obtained from applying the inverse Fourier transform
to fniy(w), iout (w), Fin (w) and
f (t)
=
f
f(w) ewtdw
out (w), respectively. The input force Fin (t) is considered as
a redundant measurement; it is used to verify the condition of quasi-static equilibrium of a
dynamically loaded specimen
Fin (t) - Fout (t) ~ 0
(2.19)
The spatial average of the logarithmic axial strain within the specimen reads
e (t)
ln
I+
u"ut
"
t
Lo
(t))
(2.20)
where LO denotes the initial length of the specimen. Using the output force measurement, we
calculate the true stress
(2.21)
o- (t) = Fout (t) exp [2v6 (t)]
AO
where AO is the initial cross-sectional area, and v is the elastic Poisson's ratio. In the present
work, it is assumed that the Poisson's ratio is constant, i.e. it depends neither on the strain
nor on the strain rate. The final true stress-strain curve is then found from the combination of
the stress and strain history functions
o' (e) = o- (t)
2.3
0 E (t)
(2.22)
Experimental Results
All experiments are performed on polyurea. This rubber-like material has a mass density of
lOg/cm 3 and an elastic modulus of about 100MPa.
2.3.1
Experiments using the Universal Testing Machine
Representative stress-strain curves for true strain rates of up to 10/s are determined from
experiments on the universal testing machine using cylindrical specimens of diameter Do =
10mm and length Lo = 10mm. All experiments are carried out under displacement control up
to a maximum true compressive strain of 1.0 (which corresponds to an engineering compressive
strain of 0.63). In order to achieve a constant true strain rate
ao,
a velocity profile
nitj
(t) of
exponential shape is applied to the top of the specimen
i, (t) = -LoO exp (aot)
(2.23)
The Poisson's ratio is determined from the experiments performed at true compressive strain
rates of up to 1/s. Based on the DIC measurements of the specimen diameter D = D (t), we
o
A
0.
75
- e-0.448
1
~-10 /s
2
~10 /s
~1
Linewr Fit
r ax
0.5 -
'E0.25 -
W0'I
0
-. 5
-0.5
-1
Logarithmic Axial Strain [-]
0
Figure 2-2: Identification of Poisson's ratio from the linear relationship between the logarithmic
radial and axial strains
calculate the logarithmic radial strain er,
(-)
=In Do
e,
(2.24)
where Do denotes the initial specimen diameter. The experimental data depicted in Figure 2-2
shows the linear relationship between the logarithmic radial strain and the logarithmic axial
strain. Upon evaluation of the slope, we find a Poisson's ratio of v = 0.448.
The data acquisition rate of the DIC system is limited to about 7Hz. Thus, we only use the
DIC system for the slowest experiments and make use of the actuator position measurement
(LVDT) to determine the effective axial displacement at higher strain rates. The comparison
of the LVDT readout with the DIC measurement yields an overall stiffness of the testing frame
of about 100kN/mm. The measured true stress-strain curves are shown in Figure 2-3a for
true strain rates of about 10-2/s, 10-1/s, 100/s, and 101/s. The corresponding true strain rate
versus true strain curves are depicted in Figure 2-3b. For the slowest experiment (to
a
10-2/S),
the slope of the stress-strain curve (Figure 2-3a) decreases significantly at a stress of about 0.1;
subsequently, the stress-strain curve changes its shape from concave to convex at an axial
strain of about 0.3. Due to the characteristic rubber chain locking behavior, the stress level
increases monotonically throughout the entire experiment from 6MPa at e = 0.1 to 13.5MPa
(a)
(b)
Universal Testing Machin e
-icn
M
0lI
-5 -....-
-- 10
1--
~-11/s-11/
-20_-_~100
.-
-1-10/~
l-~10%
-25
-10.2/S 0)
..1O1/3-
-20'.
-26
-1
-0.8
-06
-0.4
True Strain [-]
-0.2
0
- t1
-10 2
-1
1 ---
-0.8
-0.6
-0.4
True Strain [-I
-0.2
0
Figure 2-3: Test results from the universal testing machine: (a) True stress-strain curves, (b)
True strain versus true strain curves
at e = 1.0. For the next higher strain rate (to ~ 10-1/s), the overall stress level is about 12%
higher. Similarly, the shape of the stress-strain curve is preserved for strain rates of 100 /s and
101/s, but the stress level increases by 35% and 65%, respectively, compared to that at 10- 2 /S.
2.3.2
Experiments Using Conventional SHPB Systems
Appendix B outlines the identification of the frequency dependant coefficients c (w) and a (w)
for both the aluminum and nylon bars. The results are presented in Figure 2-4 together with
the Pochhammer-Chree solution (e.g. Graff, 1975). These experimentally obtained coefficients
are used throughout our analysis of the waves in both the conventional and the modified SHPB
systems.
Aluminum Bar System
Experiments at high strain rates are performed on the conventional aluminum SHPB system.
Cylindrical polyurea specimens with Do = 20mm and Lo = 5mm are used on the aluminum
system. Average strain rates of t ~ 3700/s and e ~ 2300/s are achieved at striker velocities
of 13m/s and 9m/s, respectively. To verify the quasi-static equilibrium throughout the experiments, both the input and output force are depicted in Figures 2-5a and 2-5b. The poor
5200
-0.2
-0.4
-0.6
-0.8
Experiment
--- Elastic Case
-1.2
40
30
20
10
Frequency [kHz]
-1
Frequency [kHz]
50
15
10
5
Frequency [kHz]
Figure 2-4: Propagation coefficient of aluminum bar (first row) and nylon bar (second row).
(a)-(c) Longitudinal wave speed and (b)-(d) Attenuation coefficient.
agreement of the force measurements for 3700/s may be read as lack of equilibrium (e.g. Aloui
et al., 2008). However, for the present experiments, this observation is attributed to the low
signal-to-noise ratio for the input force measurements. Due to the pronounced mismatch between the force amplitude of the incident wave (e.g. Finc
AALPALCALVt/
2
~ 120kN for
13m/s) and the specimen resistance (Fin = Asyca 9 pc ~ 15kN at e = 0.5), most of the incident wave is reflected at the input bar/specimen interface which ultimately results in a poor
input force measurement (see e.g. Grolleau et al. (2008) for details on the force measurement
accuracy). The incident wave exhibits some Pochhammer-Chree oscillations due to the lateral
inertia of the 40mm diameter aluminum bars. Consequently, we observe some non-monotonic
behavior in the stress-strain curves for 3700/s and 2300/s in Figure 2-6a. The overall shape of
the curve is very similar to that for static loading, but the stress level is almost three to four
times higher.
Nylon Bar System
Another set of high strain rate experiments (1200/s and 700/s) are performed using smaller
diameter specimens (Do = 10mm, Lo = 5mm ) on the nylon bar SHPB system. Recall that the
main reason for changing from aluminum to nylon bars is to increase the maximum duration of
the experiments from Texp = 472ps to 1255ps. At the same time, the use of nylon significantly
reduces the impedance mismatch between the bars and the polyurea specimen. This improves
the force measurement accuracy, notably, that of the input force. Striker velocities of 8m/s and
6m/s are needed to obtain an average strain rate of
a
~ 1200/s and
a
~ 700/s, respectively.
Higher striker velocities would cause inelastic deformation in the bars upon striker impact. On
the other hand, for a maximum loading duration of 1255ps, lower striker velocities would not
achieve the desired maximum true compressive strain of e = 1.0.
There are less signal oscillations in the nylon than in the aluminum system because of
its higher signal-to-noise ratio. Furthermore, due to the lower striker velocities and the wave
attenuation in the nylon input bar, there are less severe Pochhammer-Chree oscillations in
the incident wave signal as compared to the aluminum system (see Figure 2-55).
Therefore,
relatively smooth stress-strain curves are obtained from the dynamic experiments on the nylon
bar system (Figure 2-6a).
The exact evolution of the true strain rates as a function of the
(a)
*
~ --
~
(b)
~
""-
- Fi--
-
F
30-
30 -
Fout
30F
20 -
2L
Fi
Fout
0
10 -o1
5Blu0mAinum
SHPB
XiAin~nan 55
0
0 3700/s
k
0.2
0.1
Time [msec]
0
(c)
0
0.3
(d)
F
0.4
0.5
F
--
Fout
0 2-
0
Nylon SHPB
t 1200/s
r
0
0.3
0.2
Time [msec]
4-
o 2 -Q
0
0.1
6---
Fut
4-
2300s
|
0.2
0.4
0.6
Time [msec]
0.8
0
Nylon SHPB
e 700/s
0.5
1
Time [msec]
1.5
Figure 2-5: Comparison of forces between input bar and output bar from SHPB tests from
aluminum bar tests: (a) 3700/s, (b) 2300/s; and nylon bar tests: (c) 1200/s, (d) 700/s
onventional SHP B1(b0-1
3700
----
b702
)
-30
-50
'
_10 .-
--
-200s----100/
--- ~2300/s
~70/0ovetoalS
P
- - -~32300/
-------
~3700/
-60
-1
-0.8
-0.6
-0.22
1 6 -. 8-0.4
-. 4-. -.
True Strain [-]
0
0-1
-
-0.8
08
-0.6
-0.4
-.
04-. -0.2
True Strain [-]
0
Figure 2-6: (a) True stress-strain curves, (b) True strain rate versus true strain curves.
true strain are shown in Figure 2-6b. Unlike for the experiments on the aluminum bar system,
the true strain rate is no longer increasing in a monotonic manner. This is due to the lower
amplitude of the incident force (e.g. for Fine = AALPALCALVstr/
2
~ 7.6kN for e ~ 700/s) which
is now of the same order of magnitude as the specimen resistance (Fin = Aspco-spe ~ 2.5kN).
Thus, as the specimen resistance increases throughout the experiment, the magnitude of the
reflected wave decreases; as a result, despite the logarithmic strain definition, the engineering
strain rate no longer increases due to the decreasing interface velocity.
2.3.3
Experiments Using the Modified SHPB Systems
Experiments are performed on the modified nylon SHPB system using the hydraulic actuator
in an open mode, which is different from the conventional closed loop mode of servo hydraulic
testing machines. In this open loop mode, the user can preset the position of the inlet servo
valve. Furthermore, the initial pressure of the in-flowing fluid may be controlled. However,
the user has no active control of the actuator velocity throughout the experiment. Actuator
piston velocities of up to 5m/s may be achieved in this mode of operation. Here, we perform
experiments at 4m/s, lm/s, 0.5m/s, and 0.1m/s which resulted in average compressive strain
rates of about 1000/s, 110/s, 36/s and 10/s.
Three strain gages and one displacement measurement are taken into account (per bar) to
reconstruct the waves in either bar using the above deconvolution technique1 . The comparison of
the measured input and output force histories confirms the quasi-static equilibrium for 110/s,
36/s and 10/s (Figure 2-7).
The differences between the input and output force for 1000/s
are associated with the poor quality of the deconvolution based estimate of the input force;
the accuracy of the optical displacement measurement system decreases substantially at high
loading velocities leading to severe oscillations in the input force history. However, considering
that the higher velocity cases (Figure 2-5) show the good force agreement, the quasi-static
equilibrium can also be assumed for the strain rate of 1000/s. A significant force drop is found
at t ~ 5ms, 20ms, and 60ms for the strain rates of 110/s, 36/s and 10/s, respectively. This
force drop is due to the premature partial failure of the fixed end support of the output bar
that causes a short unloading-reloading cycle. The same force drops are also found in the stress
versus strain curve (Figure 2-8a) at strains of 0.55, 0.60 and 0.25 for strain rates of 110/s, 36/s
and 10/s, respectively.
2.3.4
Comment on the Signal Oscillations
There are two characteristic time scales associated with the experiments on the modified SHPB
system. The short time scale corresponds to the round trip time of an elastic wave traveling
through the specimen, Tpy
~_11ps. The second time scale is much longer; it is associated with
the round trip of an elastic wave traveling through the input bar, Ti" ~ 3600ps. The experiment
at an average strain rate of 1000/s remains unaffected by the large time scale as the total
duration of the experiment (Tep ~ 103ts) is still shorter than T,
at a strain rate of 110/s, the duration of the experiment (Te,
~ 3600ps. However, already
~-104ps) exceeds Ti
~ 3600ps.
As a result, the shape and amplitude of the incident wave is not only determined by the velocity
of the hydraulic actuator, but also by the leftward travelling wave that has been reflected by
the specimen. Consequently, the incident wave changes with a periodicity of Ti.
In the present
experiments, the first reflected wave is a tensile wave which reduces the initial magnitude of
the compressive incident wave. Hence, the rate of loading decreases before the rate of loading
increases again after the next period of Ti.
Therefore, this abrupt change of the loading rate
'The only exception is the input bar in the experiment at 110/s where one of the three strain gauge signals
was not properly recorded. Therefore, only two strain gages and one displacement measurement were taken into
account for that experiment.
-
F
F
4-
Is
0
0.2
~1000s|
0.4
0.6
0
0.8
2
Time [Msec]
6
4
Time [msec]
8
10
100
120
2
U-
0
5
20
10
15
Time [msec]
25
30
0
20
40
60 80
Time [msec]
Figure 2-7: Comparison of forces between input bar and output bar from the modified SHPB
tests: (a) 1000/s, (b) 110/s, (c) 36/s and (d) 10/s
()
(
Modified SHPB
- - - 11 /
~
-402
---
-3
I-
10
110/s
4
1000/s
-0.6
-
U)
10/s
__36/s
-300
-0.8
110/s
s1000/s
--10
-40
-1
Modified SHPB
/5
S636/s
-0.4
-0.2
-10
0
-1
-0.8
-0.6
-0.4
-0.2
0
True Strain[-
True Strain [-]
Figure 2-8: Test results from the modified SHPB system: (a) True stress-strain curves, (b) TRue
strain rate versus true strain curves.
has a periodicity of Tin. The corresponding strain rate versus strain curve shows a pronounced
decrease in strain rate; since the strain increases only little during a period of reduced loading
rate, we observe sharp drops in the strain rate versus total strain curve. For lower average
strain rates, this number of strain rate drops increases further. Formally, we may write
n e& Etot/&avel
(2.25)
in
where n is the number of the expected drops in strain rate associated with the wave reflections
in the input bar. This number is 2, 7, and 27 for the experiments at average strain rates of
110/s, 36/s and 10/s. In the limiting case of static loading, we have n -+ oo which ultimately
results in a constant strain-rate versus strain curve. In addition to loading velocity changes
associated with wave reflections in the input bar, our experimental results are also affected by
other sources of vibrations. These include wave reflections within the output bar as well as
vibrations in the fixed end support system and the hydraulic actuator. Therefore, the exact
identification of all strain rate drops in Figure 2-8b according to Eq. (2.25) has been omitted.
(a
-
(b)
0
-10
-- 10--
-20
-
---
--
-~
-1
-- ~1200/s SHPB
-
-----
-
-0.8
2
-
-10
iz~
-30
-10
Universal Testing Machine
10/s Hydraulic SHPB
~ 1000/s Hydraulic SHPB
10/s Universal Testing Machine
10/s Hydraulic SHPB
1000/s Hydrauli SHPB
1200/s SHPB
-0.6
-0.4
True Strain [-]
-0.2
-3
10
--4
0
-1
-0.8
-0.6
-0.4
True Strain [-]
-0.2
0
Figure 2-9: Comparison of the results from the modified SHPB with those from other two
testing methods: (a) True stress-strain curves, (b) True strain rate versus true strain curves.
Discussion
2.4
Experimental Results
2.4.1
To validate our experimental data, we first checked the consistency among the results obtained
from different testing methods. Figure 2-9 shows selected stress-strain curves obtained from the
modified SHPB system (dashed lines) next to the results from the conventional SHPB (red solid
line) and the universal testing machine (blue solid line). For 1000/s, the modified SHPB result
shows reasonably good agreement with the conventional SHPB curve for 1200/s. Analogously,
for the average strain rate of 10/s, the stress-strain curve obtained from the modified SHPB
test corresponds well to that obtained from the test on the universal testing machine. Recall
that the perturbation of the stress-strain curve for the modified SHPB system at about 0.25
strain is due to the partial premature failure of the output bar end support system. The stress
level from the modified SHPB is slightly higher after partial support failure which is attributed
to differences in the strain rate.
The data in Figure 2-10a show the stress as a function of the strain rate for different levels
of strain: 0.1, 0.5 and 0.9. The effect of strain rate is more pronounced at large strains. For
instance, at a strain of 0.1, the stress level increases by 317% when increasing the strain rate
from
t
~ 10- 2 /s to & ~ 3700/s (increase 6kN from to 19kN); at a strain of 0.9, the stress level
I
I
I
to
A.
0
A
00
-J
2l
0)
(-
-400
/
+
at e= -0.1
o
at
&
at
ed at e = -04
Blue at Et= -0.9
a
-0.5
Etr=
-103 -102 -10
-10
True Strain Rate
- 10
-10
O0
A
-0.9
I
_1 1 I I
-10
__5
-10
-10
-
R
asec1
-10
3
ff
-10
1
Roland et al. (2007)
Sarva et al. (2007)
Current 'esearch
-10
True Strain Rate [Isec]
-103
Figure 2-10: True stress as a function of the strain rate at selected strain levels: (a) Results of
the current study and a fit of Eq. (2.26) to the results from the present study. (b) Comparison
of the results with previous studies.
increases by 360% over the same range of strain rates (from 12.5kN at
a
a
~ 10-
2
/s to 45kN at
~ 3700/s). Using the separable form of strain and strain-rate effect, in the same figure, the
following nonlinear empirical function has been fitted to the experimental data:
o,
f (e)g(W)
= (-)
where
f (e)
1+ A
(2.26)
aref
B
is the asymptotic stress-strain curve at infinitely slow loading conditions and g
represents the contribution of strain rate sensitivity. Here, the reference strain rate of
(e)
aref
2
10- /s is chosen to be the lowest strain rate in the present experiments. In addition, A = 0.0899
is responsible for the amplitude of the strain rate contribution, and B = 0.249 represents the
sensitivity of the strain rate.
2.4.2
Intermediate Strain Rate Testing Systems
The present experimental study confirms the high strain rate sensitivity of the polyurea material
which has been reported in earlier studies. Roland et al. (2007) performed a series of tensile
tests using a custom-made pulley system in a drop tower to perform uniaxial tensile tests at
intermediate and high strain rates.
Sarva et al.
(2007) used an enhanced universal testing
machine to perform compression experiments at strain rates of up to 80/s while an aluminum
SHPB system with a striker bar length of 3m has been used to perform experiments at strain
rates above 800/s. The comparison of the present experimental data with the results of Sarva
et al.
(2007) and Roland et al.
(2007) confirms the validity of the measurements with the
modified SHPB system (Figure 2-10b).
The implementation of the deconvolution technique by Bussac et al. (2002) leads to a stable
algorithm that is convenient to use for the reconstruction of dispersive waves in bars based
on redundant measurements. Thus, the theoretical limitation of the duration of experiments
on SHPB systems is successfully overcome.
In combination with a hydraulic actuator, the
entire range of low to high strain rates could be covered using a single testing system. The
comparison with conventional SHPB experiments at high strain rates and universal testing
machine experiments at low strain rates has confirmed the validity of the modified SHPB
technique. However, there are still two difficulties associated with our modified SHPB system
which need to be addressed in the future:
9 Displacement and/or velocity measurement accuracy: the accuracy of the deconvolution
technique relies heavily on accurate displacement measurements (in particular at low
strain rates). The present optical technique provided good results for loading velocities
of up to 0.5m/s, but significant errors became visible at larger loading velocities.
e Quality of the loading pulse at specimen interfaces. In order to achieve approximately constant strain rates, the ideal loading pulse should be such that the bar/specimen interfaces
move at constant velocities.
The first difficulty may be resolved through the use of improved measurement equipment.
Alternatively, the deconvolution technique for high loading velocities may also be applied using
strain gage measurements only. However, it is very challenging to overcome the second difficulty.
As an intermediate strain rate experiment takes much longer than a wave round trip in the
input bar, the input bar/specimen interface velocity is not constant even if the hydraulic piston
moves at a constant velocity. Simple wave analysis shows that a period of high velocity loading
is followed by a period of loading at a lower rate; the length of each period corresponds to the
round trip time for a wave travelling in the input bar. The same holds true for the output
bar/specimen interface velocity which is affected by the round trip time in the output bar.
Consequently, the strain rate in our intermediate strain rate experiments was not constant.
For the desired maximum true compressive strain of e = 1.0, the total duration of an
intermediate strain experiment at 50/s is Texp = 20ms - irrespective of the specimen geometry.
Conceptually, there exist several solutions to this problem:
* Conventional nylon SHPB system with a striker bar length of 1740 x 0.02 x 0.5 = 17.4m
along with a 35m long input bar and a 17.5m long output bar. In this configuration, all
strain gages can be positioned such that the rightward and leftward traveling waves do
not superpose at the strain gage locations.
" Conventional nylon SHPB system with a 17.5m long striker bar and 17.5m long input
and output bars. In this case, a deconvolution method needs to be used to reconstruct
the waves in the input and output bars. However, the input bar is still sufficiently long
to guarantee that the round trip time is greater than the duration of the experiment.
" Hydraulic nylon SHPB system with 17.5m long input and output bars. Based on the
assumption that the hydraulic piston moves at a constant velocity, this system will provide
the same capabilities as the previous system.
As an alternative to very long input and output bars, one may chose the opposite strategy.
Note that the magnitude of the oscillations is proportional to the change in force level in
the specimen over the time Ti,. Thus, the shorter the input bar, the smaller the oscillation
magnitude. One could therefore envision very short (e.g. < 0.5m) small diameter input and
output bars. In this case, we have Tin = 0.57ms and hence Ti
<
Tep.
However, since the modified SHPB system requires two displacement measurement sensors
(notably for low strain rate experiments), one can also use these sensors to measure directly
the displacements of the respective bar/specimen interfaces. Hence the strain history can be
measured without using the deconvolution algorithm. The bars would therefore only serve as
load cell to measure the force history. Unless the quasi-static equilibrium needs to be verified
experimentally, a single force measurement is satisfactory. Moreover, it may be worth considering a piezoelectric sensor to measure the force, thereby completely eliminating the use of bars
to perform the experiments at low, intermediate and high strain rates. As our hydraulic piston
cannot provide a constant loading velocity above 0.5m/s, a striker bar may also be used to load
the specimen. The only unknown which is left in this system is the realization of the "fixed"
boundary condition. Further research needs to be carried out to design a support point that
does not introduce spurious oscillation into the testing system.
2.5
Conclusion
The modified SHPB system of Zhao and Gary (1997) has been used to perform compression tests
on polyurea at low, intermediate and high strain rates. It is composed of nylon input and output
bars, while the striker bar is substituted by a hydraulic actuator.
Using the deconvolution
technique by Bussac et al. (2002), the time limitation of conventional SHPB systems may be
overcome, thereby enabling the use of the modified SHPB system for low and intermediate strain
rate experiments of long duration. The experiments confirm the known strain rate sensitivity
of polyurea. The measured stress levels correspond well to earlier results which have been
obtained from tests on conventional SHPB systems with very long bars. Although the intrinsic
time limitation of SHPB systems could be overcome, this study also shows that it is still not
possible to perform experiments at reasonably constant strain rates with this technique. This is
due to the finite length of the input and output bars which causes a periodic change in loading
velocity. It is shown that intermediate strain rate SHPB experiments require either very long
bars (> 20m) or very short bars (< 0.5m) in order to achieve an approximately constant strain
rate throughout the entire experiment.
Chapter 3
Constitutive Modeling
3.1
Introduction
Polyurea is used to mitigate structural damage during impact loading because of its good
damping performance. In addition, it is utilized by various industries because of its fast setting
time as well as its good chemical and fire resistance. Polyurea has found applications in army
vehicles for blast protection because of its high toughness-to-density ratio at high strain rates.
Polyurea shows a highly nonlinear viscoelastic behavior at finite strains (e.g.
Amirkhizi et
al., 2006, Bogoslovov and Roland, 2007, Roland et al., 2007, Shim and Mohr, 2009a). The
mechanical properties of linearly viscoelastic materials may be described by the relaxation
modulus (or creep compliance) which is independent of strain magnitude. However, nonlinear
viscoelasticity is characterized by a decrease (or increase) of the relaxation modulus (or creep
compliance) with increasing strain or decreasing stress (e.g. Brinson and Brinson, 2008).
Most finite viscoelasticity models of elastomers are formulated using either (1) the so-called
hereditary integral approach or (2) the framework of multiplicative decomposition of the deformation gradient.
Motivated by linear viscoelastic models, hereditary integral models are
formulated in terms of relaxation or memory functions (e.g. Lockett, 1972). Widely used single
integral theories are the theory of Finite Linear Viscoelasticity (Coleman and Noll, 1961) and
the BKZ theory (Bernstein et al., 1963); both make use of several relaxation/memory functions.
Significant efforts have been made to improve these theories and to reduce the number of required material parameters (e.g. Lianis, 1963, McGuirt and Lianis, 1970, Leonov, 1976, Johnson
et al., 1994, Haupt and Lion, 2002). Nonlinear viscoelastic behavior is often considered as the
superposition of a rate-independent nonlinear elastic response (so-called equilibrium part) and a
viscosity-induced overstress contribution which is described through fading memory functions.
Most nonlinear viscoelastic constitutive models have been experimentally validated at very low
strain-rates of 0.1/s or less. Only few papers deal with the nonlinear viscoelastic behavior of
elastomers at intermediate and high strain-rates (1/s to 1000/s). Using the hereditary integral
approach, Yang et al. (2000) and Shim et al. (2004) proposed a phenomenological constitutive
model to predict the behavior of silicon rubber at high strain rates (900/s to 3000/s). Hoo Fatt
and Ouyang (2007) adopted the integral approach to model the response of butadiene rubber at
strain rate ranging from 76/s to 450/s. The validity of most hereditary integral approach based
models is limited to a narrow range of strain rates due to the use of only one relaxation time
period (Yang et al., 2000 and Shim et al., 2004) or a constant memory function (Hoo Fatt and
Ouyang, 2007). In general, the hereditary integral approach is very useful in describing finite
viscoelastic behavior, but its successful application to the real test data depends strongly on
the effectiveness of the rather complex relaxation or memory function calibration procedures.
Compared to hereditary integral models, the multiplicative decomposition of the deformation gradient typically leads to models with material parameters that can be easily identified
from experiments. The concept of the multiplicative decomposition of the deformation gradient (Kroner, 1960, and Lee, 1969) was initially applied to finite viscoelasticity by Sidoroff
(1974) and further explored by others (e.g. Lubliner, 1985, Le Tallec, 1993, Reese and Govindjee, 1998, Huber and Tsakmakis, 2000). The nonlinear viscoelasticity is commonly described
through a rheological spring-dashpot model that features a rate-independent equilibrium part
and a rate-dependent viscous part. In particular, Zener models are widely used (e.g. Roland,
1989, Johnson et al., 1995, Bergstrom and Boyce, 1998, Huber and Tsakmakis, 2000, Quintavalla and Johnson, 2004, Bergstrom and Hilbert, 2005, Qi and Boyce, 2005, Tomita et al.,
2008, Areias and Matous, 2008).
Zener models are composed of a spring (rate-independent
part) in parallel with a Maxwell element (rate-dependent part). The constitutive model of the
time-dependent part includes an evolution equation for the viscous deformation. The simplest
evolution equation is a linear viscous flow rule (e.g. Amin et al., 2002, Johlitz et al., 2007).
However, experiments have shown that the viscosity at finite strains is a function of the driving
stress (e.g. Amin et al., 2006), the strain (e.g. Amin et al., 2006, Hoo Fatt and Ouyang, 2008)
and/or the strain-rate (e.g. Khan et al., 2006). Thus, nonlinear viscous flows rules have been
developed using a micro-mechanism inspired approach (e.g. Bergstrom and Boyce, 1998, Palm
et al., 2006) or a purely phenomenological approach (e.g. Khan and Zhang, 2001, Colak, 2005,
Amin et al., 2006, Khan et al., 2006, Hoo Fatt and Ouyang, 2008).
In order to represent the equilibrium contribution as the nonlinear elastic spring, rubber
elasticity models are employed using the conventional material types such as Neo-Hookean (e.g.
Johlitz et al., 2007), Mooney-Rivlin, (e.g. Huber and Tsakmakis, 2000), or Arruda and Boyce's
(1993) eight-chain model (e.g. Bergstrom and Boyce, 1998, Tomita et al., 2008). Other research
groups (e.g. Amin et al., 2002, 2006, Hoo Fatt and Ouyang, 2008) used modified conventional
material models to describe the high initial stiffness and the subsequent strain hardening.
Although many researcher have recently reported on experimental behavior of various polymeric materials for a wide range of strain rates (e.g. Khan and Farrokh, 2006, Mulliken and
Boyce, 2006, Sarva et al., 2007), little research has been reported on the modeling of elastomeric
materials for loading and unloading for a wide range of strain rates. As for the hereditary integral approach based models, most multiplicative decomposition based models have been experimentally validated for a narrow range of strain rates. Quintavalla and Johnson (2004) adopted
the Bergstrom-Boyce model to describe the dynamic behavior of cis-(1,4) polybutadiene at high
strain-rate of 3000/s to 5000/s. Recently, Hoo Fatt and Ouyang (2008) proposed a thermodynamically consistent constitutive model with modified neo-Hookean rubber elastic springs
to describe the steep initial stiffness of virgin butadiene rubber under tensile and compressive
loading at intermediate strain rates (76/s to 450/s).
The present work focuses on the modeling of the loading and unloading response of polyurea
over a wide range of strain rates (from 10-/s to 101 /s) and at large compressive strains (up
to 1.0). Using the framework of the multiplicative decomposition of the deformation gradient,
the rheological concept of two parallel Maxwell elements is employed to develop a nonlinear
viscoelastic finite strain constitutive model. Guided by similar developments in area of ratedependent plasticity models for glassy polymers (e.g.
Qi and Boyce, 2005, Anand and Ames,
2006, Anand et al., 2009, Ames et al., 2009, Ayoub et al., 2009), the high stiffness of elastomers
at small deformations is modeled using Hencky's strain energy function. In view of the simula-
tion of blast and impact events, we limit our attention to the mechanical behavior of polyurea
in its virgin state. After presenting all experimental results in Section 3.2, the constitutive
model is detailed in Sections 3.3 and 3.4. A discussion of the comparison of simulations and
experimental results for polyurea is given in Section 3.5.
3.2
3.2.1
Experimental Investigation
Material
The polyurea specimens used in the study are extracted from a 12.7mm thick polyurea DragonShieldHT Explosive Resistant Coating (ERC) layer on a steel armor plate. After separation from the
plates, the polyurea specimens of diameter Do = 10mm and length LO = 10mm are machined
using conventional milling procedures. The shelf life of the polyurea prior to testing was about
4 years. All experiments are performed on polyurea in its virgin state (condition after curing.
no prior mechanical loading history).
3.2.2
Relaxation Experiments
The nonlinearity of the viscoelastic behavior of polyurea at low and intermediate strain rates
is identified from relaxation tests. Using a hydraulic universal testing machine (Model 8800,
Instron), the material is rapidly loaded to a prescribed strain level at the beginning of the
experiment. The strain is subsequently kept constant while stress relaxation takes place. Nine
relaxation tests are performed for true compressive strains ranging from 6o = 0.048 to 1.0. The
engineering strain rate during the rapid loading phase was about 0.48/s. In order to capture
the short-term viscoelastic behavior, the sampling frequency of the data was 5kHz; the stress
is recorded for about 10 minutes to identify the long-term behavior. Figure 3-la shows the
recorded stress histories for various constant strain loadings. While the test duration was not
long enough to observe the asymptotic long-term behavior (i.e.
equilibrium modulus), the
results indicate that the sampling frequency of 5kHz is high enough to capture the asymptotic
short-term viscoelastic behavior (i.e. initial relaxation modulus).
---
,
-e
= -0048
= -019
--
= -0.29
,=-0.40
r,
eo - -0.53
-
-e
= -0.67
=-083
--
10
-A
10-
3
-2
10-
Time [sec]
'i~100
-0.048
--
0
-1
10 10 10
Time [sec]
1
10
2
3
10
100
-0 if
NI
.
........
-019
NI
-029
-
-0.40
0
i
60
.
.
-
-0.63
C0
-eo=-083
0
-
ifeo -1.0
-0
10
10
Time [sec]
-32"
"
0
10-2 10 101 10
Time [sec]
""
2"
102 103
-i--,= 616 sec
WU
W0
P4
F)
t6--VO'sec
-
- ,=10-'sec
S-=10
-
0
0
sec
see
-15
a-
-
-.
-20.11
-1
-0.8 -0.6 -0.4 -0.2
True Strain [-]
Figure 3-1: Results of relaxation tests (left column) and corresponding simulations (right column). (a)-(b) True stress histories, (c)-(d) Relaxation moduli histories, (e)-(f) Isochronous
stress-strain curves for different instants after the rapid strain loading.
The relaxation moduli are computed using the definition
H(t) =
o-(t)
(3.1)
where -(t) is the time history of the measured true stress for a constant strain of sO (Figure
3-la). The corresponding relaxation modulus histories (Figure 3-1c) clearly demonstrate the
nonlinear viscoelastic nature of the material response: the relaxation modulus decreases from
120MPa (at the strain of -0.048)
to 20MPa (at the strain of -1.0)
at the beginning of
relaxation; after more than 10 minutes, the corresponding relaxation moduli are 60MPa and
1OMPa, respectively. The strong nonlinear nature of the viscoelastic behavior becomes also
apparent when plotting the isochronous stress-strain curves (Figure 3-le).
3.2.3
Continuous Compression Experiments
In addition to relaxation experiments, constant strain rate loading and unloading tests are
performed at absolute true strain rates between 10-
and 10 1 /s (Figure 3-2a). The measured
true stress versus logarithmic strain curves for five different strain rates are summarized in
Figure 3-2b. At least two tests are performed at each strain rate to confirm the repeatability
of the experimental results. During the loading phase, the observed stress level exhibits strong
3
strain rate sensitivity. For example, the observed stress level for 10- /s is about twice as high
as for 101 /s.
However, during unloading, all the stress-strain curves seem to converge to the
same curve regardless of the unloading strain rate.
3.2.4
Stair Compression Experiments
In order to investigate the stress relaxation behavior during loading and unloading, experiments
are performed with stair-type strain history profiles. Figure 3-3a shows the applied strain history
as a function of the normalized time t k/Eol. The same maximum strain of
for all experiments. Experiments are performed for
lEoI
= 10
3
Co
= -1.0
is chosen
/s, 10 2 /s, 10-/s, 10a/s and
101 /s. All measured stress-strain curves are depicted in Figure 3-3b. The results from the stair
compression tests confirm the previous observation of high strain rate sensitivity during loading
phase along with nearly rate independent behavior during unloading. Also the magnitude of
(a)
'
'
-0.2---- 0.4 (D
S-0.6 -
-0.8
-1
0
(b)
I
I
0.5
1
Normalized Time,
0
2
1.5
tio/eol
(c)
-5 -
-- 1
I
[-1
0
-5 -
/
<-10-
-15 -
-15 -
~~
~1r/ec
4
_
-20 -
----
-25
~ ~ 10
/sec
10J/sec
~e1/sec
-1
io ec
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
10- /Sec
2
10 /sec
-20
2,, -
10 /sec
10 /sec
-25.10'/s
-1
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
Figure 3-2: Results of continuous loading/unloading experiments and corresponding simulations. (a) History of applied strain. Note that time in the x-axis is normalized by the total
testing duration Ieo/o| where so = 10-3 , 10-2, 10-1, 100, and 101 /s . (b) Stress-strain curves
obtained from experiments and
stress relaxation during the loading phase is found to be larger than that during unlading. The
superposition of the results from the monotonic and stair compression experiments (Figure 33d) shows that the specimens recover the virgin state resistance (described by the curves from
the monotonic tests) after each relaxation period in the stair compression experiment. The
resistance recovery appears to be slower for large strains and high strain rates.
3.3
3.3.1
Constitutive Model
Motivation
In the framework of multiplicative decomposition of the deformation gradient, models of the
Zener type (Maxwell element in parallel with a spring) are widely used since these can effectively represent the nonlinear viscoelastic behaviors of many elastomers.
For Zener models,
it is common to identify the equilibrium path and the over-stress contribution from the experimental stress-strain curves. The equilibrium path can be determined in an approximate
manner from two alternative approaches: (1) taking the mid-path of stress-strain curves from
the monotonic loading and unloading test (e.g. Bergstrum and Boyce, 1996), (2) connecting
the stress relaxation mid-points between loading and unloading phase from the stair-type compression tests (e.g. Qi and Boyce, 2005).
Figure 3-4 shows the equilibrium paths that have
been identified from the continuous compression tests at different strain rates. The summary
plot in Figure 3-66a clearly shows that the identified equilibrium paths are not identical, i.e.
the identified equilibrium path depends on the strain rate. For instance, at a strain of -1.0,
an equilibrium stress of -1OMPa is found from the experiment at 10stress of -17MPa is obtained from the experiment at 101 /s.
3
/s while an equilibrium
This important observation is
confirmed by the stair-type compression tests. Recall from Figure 3-33b that the amount of
stress relaxation is different for loading and unloading. Thus, it is concluded that the concept
of equilibrium path breaks down in the case of polyurea. Consequently, rheological models of
the Zener type will not be able to describe the nonlinear viscoelastic behavior of polyurea for
both loading and unloading.
In models of the Zener type, the equilibrium part is represented by a spring element.
Qi and
Boyce (2005) explained in their paper on the micro-mechanism inspired constitutive modeling
(a)
0
(b)
2
4
6
8
Normalized Time, t~go/eol [-]
u
-5
-10
i-10
~-15
T-15
0)
6-
-20
-20
-
,
-
ItTsec
/0
-
-25
-1
True Strain [-]
(d)
10*/sec
10' /sec
-0.8 -0.6 -0.4 -0.2
True Strain [-]
(e)
0
-5
-10
-15
I-
True Strain [-]
True Strain [-]
Figure 3-3: Results for stair compression loading/unloading. (a) History of applied strain.
Note that time in the x-axis is normalized by the total testing duration leo/sol where co =
1 and to = 10-3 , 10-2, 10-1, 100 , and 101 /s . (b) Stress-strain curves from experiments
and (c) Corresponding simulations. (d) Comparison of results from continuous and stair-type
loading/unloading experiments and (e) Corresponding simulations.
(a)
'
'
--
---
'
/
(b)
-
-5 --
-10 -
10 -
-15-
-O-15-
-20-
-20-
-
- Test
Median
3 Approximate
CurvefitwithGentMode- -
-25
-0.8 -0.6 -0.4
True Strain [-]
-1
-0.2
-25
0
(c)
(d)
--0
-1
Test
Median
Approximate
Curvefi withGentMode
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
0
-5-5-10
I 2r
c,
4-)
-15-
(n-15-
a)
-20-25
(e)
- - - Test
Median
0 Approximate
CurvefiteitnGentMode)
'
-0.8 -0.6 -0.4 -0.2
True Strain [-]
-1
0
0
a
-20-
-0
-251
-
-1
-0.8
- Test
Median
Approximate
Curvefitwth GentMode
-0.6 -0.4 -0.2
True Strain [-]
0
-
-5
)-105--
-20 -
i
-25
'
-1
|
3
---
Test
Approximate Median
Curvefg wih Gent Modei
-0.8 -0.6 -0.4
True Strain [-]
-0.2
0
Figure 3-4: Estimation of the behavior of Network A using the equilibrium path concept for
path concept for different strain rates: (a) 10-3/.S, (b) 10-2/s, (C) 10-1/s, (d) 100/s and (e)
101/s
Nonlinear
Linear
Viscous Damper
Elastic Spring
Network B
Network A
Nonlinear
Nonlinear
Viscous Damper
Rubber Spring
Figure 3-5: Proposed rheological model for polyurea composed of two Maxwell elements in
parallel.
of thermoplastic polyurethanes that the spring element associated with the equilibrium path
represents the entropic resistance of the soft domains in polyurethanes, while the rate dependent
behavior of the hard part is represented by the Maxwell element of the Zener model. Due to
the apparent rate dependency of the calculated equilibrium paths, we substitute the spring of
the Zener model by another Maxwell element. The resulting rheological model, two Maxwell
springs in parallel (Figure 3-5), has already been considered in the past by Boyce et al. (2000)
and Dupaix and Boyce (2007) to describe the finite strain behavior of amorphous polymers. In
the context of amorphous polymers near glass transition temperature, two Maxwell elements
have been used to represent the rate-dependent inter- and intra-molecular network resistances
under monotonic loading conditions.
In the following, we outline the constitutive equations for each of the four basic elements
of our rheological model. Throughout our presentation, we refer to the Maxwell element that
is predominantly associated with the deformation resistance of the soft domain of polyurea
as "Network A", while the Maxwell element associated with the effect of the hard domain is
referred to as "Network B".
3.3.2
Homogenization
The macroscopic deformation gradient Ftat is decomposed into a rotation free volumetric part
FvOl = J1 i
3
1
with
J = det Ft0 t
(3.2)
and an isochoric part F
F = J-1/3 Feto
(3.3)
The constitutive model for the stresses induced by isochoric deformation is developed with
the Taylor assumption in mind, i.e. the isochoric deformation gradients within both networks,
FA and FB, equal the macroscopic isochoric deformation gradient
(3.4)
F = FA= FB
while the deviatoric part of the macroscopic Cauchy stress T corresponds to the spatial average
of the local stress fields
devT =
f
-dQA +
adQBI
(3.5)
where o- describes the local deviatoric stress fields, Q is the total volume of the representative volume element, and QA and QB are the corresponding volumes of Networks A and B,
respectively. In the context of micromechanism-inspired polymer models, the role of the microstructure is typically neglected. Instead, the constitutive equations for the individual phases
predict a weighted macroscopic Cauchy stress (in the present case the weighted deviator stresses
TA and TB) such that the macroscopic stress deviator can be written as
devT = TA + TB
3.3.3
(3.6)
Constitutive Equations for Volumetric Deformation
The volume change of polyurea under mechanical loading is very small as compared to the
magnitude of isochoric deformation.
For simplicity, we assume a linear elastic relationship
between the logarithmic volumetric strain and the hydrostatic part of the macroscopic Cauchy
stress tensor
trT= n
3
J
where r, denotes the bulk modulus.
(3.7)
3.3.4
Maxwell Model for Isochoric Deformation
The responses of Network A and Network B are described through Maxwell models. In this
subsection, we provide the general framework for the formulation of the constitutive equations
for Maxwell models. In the Sections 3.3.5 and 3.3.6, we will then specialize our equations for
the individual networks.
Kinematics
Consider a Maxwell element K subject to a isochoric deformation gradient FK with det FK = 1.
In the context of finite strain, we assume that the total isochoric deformation gradient can be
multiplicatively decomposed into an elastic part F'- and a viscous part F',
(3.8)
FK = FegFv
Furthermore, we introduce the total, elastic and viscous rate of isochoric deformation tensors
FK = LKFK
, PK
=
LK F
and
Y-
= Lv Fv
(3.9)
with the relationship
LK = Le< + Fe L Fe-1
(3.10)
It is assumed that both the elastic and viscous spins are zero for the deformation of the Maxwell
elements. Formally, we write
D' := L' = LT and Dv := Lv = LvT
(3.11)
Due to the spin-free elastic deformation, the rate of deformation tensor D" may also be expressed as a function of the time derivative of the right Cauchy-Green tensor
D = 1FV-Tae Fe-1
(3.12)
In addition to the tensor description of the kinematics, we make use of the scalar deformation
measure
(3.13)
(= vtrC- 3
in our model formulation. The strain-like variable ( is zero in the undeformed configuration
and always positive for deformed configurations.
Thermodynamics and Hyperelasticity
For isothermal conditions, the second law of thermodynamics for each Maxwell element reads
JTK : LK
-
K
(3.14)
0
with the deviatoric Cauchy stress TK and the free energy OK (per unit initial volume). We
impose the assumption of Green elasticity through the hyperelastic relationship (e.g. Ogden,
1984)
TK =-dev F
JL&CK
where C
K FeT
(3.15)
= FTFe denotes the right Cauchy-Green tensor of isochoric elastic deformation.
Recall that we assume isochoric elastic deformation which implies det F' = 1 and trD< = 0.
Using Eq. (3.10) and Eq. (3.15) in Eq. (3.14) yields the thermodynamic constraint
JTKe:
(FeLF~j1 )
(JFTTKF-T)
Dv< > 0
(3.16)
In the following, the Mandel stress
MK:= Jdev
{FK TKF}
(3.17)
is used as the driving stress of viscous deformation while the viscoplastic constitutive equations
are written such that
MK: Dv > 0
(3.18)
Flow Rule for Viscous Flow
The direction of viscous flow is supposed to be aligned with the direction of the driving Mandel
stress. Formally, we write the flow rule
(3.19)
MK =I7KDI
with the deformation and rate dependent viscosity riK > 0. The rate dependent viscosity is
defined through the nonlinear relationship nk =
ffk
(Jk) between the equivalent Mandel stress
fnk
(3.20)
-Mk- Mk
2
and the equivalent viscous strain rate
dk
D
(3.21)
Using Eq. (3.20) and Eq. (3.21) in Eq. (3.19), we obtain
(3.22)
7K-2 rnk
3.3.5
Specialization of the Maxwell Model for Network A
Network A represents the rubbery response of the soft domain which is modeled through a
Maxwell element composed of a nonlinear elastic Gent spring and a nonlinear viscous damper
that accounts for the apparent rate dependency of the theoretical equilibrium path.
Gent Elasticity of Network A
For isochoric deformation, Gent's (1996) free energy function may be written as
1
A
/
2
AJA in
_
trC' - 3
A
JA
(3.23)
with the material parameters PA
0 (initial modulus) and
>
JA
> 0 (locking stretch). With this
form of free energy, the Cauchy stress TA in Network A reads:
p
TA
1-
trCe - 3A
J
1
(3.24)
devBe
A
JA
where B' = F FT denotes the right Cauchy-Green tensor of isochoric elastic deformation.
Nonlinear Viscous Response of Network A
Figure 3-6a suggests that the viscoelastic contribution to the stress of Network A increases
monotonically as the deformation increases. Furthermore, the experimental results suggest a
power-law relation between the viscosity r/A and the equivalent rate of viscous deformation.
Thus, the nonlinear viscous evolution law for Network A is written as
3
-n
MA = -r7AdA
2
= PA (exp ( - 1)
^^
dref
(3.25)
with the reference rate of deformation dref = 1/s, the viscosity constant PA > 0, and the
exponent nA > 0 controlling the rate-sensitivity.
3.3.6
Specialization of the Constitutive Equations for Network B
Network B represents the response of the hard domain which is modeled through a Maxwell
element composed of a linear elastic spring and a nonlinear viscous damper that accounts for
the high stiffness at small strain and the rate sensitivity of hysteresis.
Hencky Elasticity of Network B
The stiffness of Network B governs the stiff macroscopic response of polyurea at small strains.
Here, we make use of the isochoric part of Hencky's strain energy function
B =
1
: (ln U)B
2B B (lnUe)
(3.26)
where the elastic stretch tensor U' is found from the polar decomposition of the elastic deformation gradient, Fe = R'Ue with RfR' = 1. Upon evaluation, we obtain the deviatoric
part of the Cauchy stress acting on Network B
TB= 2 PBRB (lnU')R eT
(3.27)
Nonlinear Viscous Response of Network B
Figure 3-7a suggests that the overall viscoelastic response of Network B is similar to the linear
viscoelastic response of a Maxwell model composed of a linear spring (of Young's modulus E)
and a linear viscous damper (of viscosity r). The stress-strain (o- - e) response of a Maxwell
model with the linear components at a constant strain rate reads (e.g. Brinson and Brinson,
2008)
o-= ETO I - exp
where
T
= r//E is the relaxation time, and
so
(.
(3.28)
is the applied constant strain rate. Note that
even though the Maxwell model is composed of only linear elements, the stress-strain response
is nonlinear at constant strain rates. However, our experiments suggest that the stress-strain
response of Network B is linear for small strains regardless of strain rate.
Inspired by the results for the one-dimensional Maxwell model with linear components
(Eq. (3.28)), we propose a constitutive equation that guarantees a linear elastic response of
Network B at small strains
rnB =
V 3 BQB
dBK're 1- exp
(dref )
[BQB
Q
(3.29)
dre
dref
I
where the dimensionless material constants QB and nB characterize the viscous properties of
Network B.
3.4
Identification of the Model Parameters
The proposed constitutive model comprises eight material parameters: four parameters (PA, JA,
PA, nA) for Network A, three parameters (pB, QB, nB) for Network B, and one parameter (r,)
describing the elastic volumetric response. The constitutive equations are detailed for uniaxial
stress loading in order to identify all material parameters from the monotonic loading and
unloading compression experiments.
3.4.1
Constitutive Equations for Uniaxial Loading
Under uniaxial stress loading, the macroscopic stress tensor takes the form
(3.30)
T = o-ei 0 ei
where o- is the true stress (macroscopic Cauchy stress) while the macroscopic deformation
gradient reads
Ftat = Aiei & ei + A2 (e2 0 e 2 + e3 @0e 3 )
(3.31)
J = det Ftot = A1 (A2 ) 2
(3.32)
At the network level, the isochoric part of the deformation gradient may be expressed as a
function of the stretch A,
1
(e 2 9 e2 + e3 0 e3 ) with A
VA
F = Aei 0 ei +
-)
2 3
/
(3.33)
The total stretch of a Maxwell element K can be decomposed further into its elastic part A'K
and the viscous part A'K
A'
(3.34)
A'
Using this notation, the kinematic variables used in our constitutive equations may be
expressed as a function of the elastic and viscous stretches and their respective time derivatives
(3.35)
trCe = (Ae )2 + 4
K
deB =- I
deBK=3
In U
eI
(Ae )2 K
Ae
K
AK
(2ei 0 ei - e2 09e2
1
(InAg) (2ei 0 ei
2
= -
A2+
-
- e3
o e 3)
e 2 0 e2 - e 3 0 e 3 )
- 3
(3.36)
(3.37)
(3.38)
AK
d
e=
(3.39)
,
The deviatoric stress tensors TK acting on the Maxwell elements are written as
TK=
3
(2e1
- e2 & 2 - e3 0e
(3.40)
3)
while
C- =
JA
and
c-B
(3.41)
OA + OrB
are true stresses acting on Networks A and B, respectively.
Using expressions Eq. (3.30) to Eq. (3.41), we obtain the simplified constitutive equations
for uniaxial loading:
e Volumetric deformation induced linear elasticity with the model parameter r,
ln J
(3.42)
" Gent elasticity of Network A with the model parameters
JA
and PA
Ke2 + +-1
o A(Ai)2_
(3.43)
* Nonlinear viscous response of Network A with the model parameters PA and nA
o-AUl = PA (exp
J
- 1)
re
)
" Hencky elasticity of Network B with the model parameter
O-B
(3.44)
drefI
1
B
(3.45)
n AB
" Nonlinear viscous response of Network B with the model parameters and
BQB
IO'I V3J JBI
deB )n(B
dref
eXp
1-ex
dB (Yf
QB
\dref
/
(3.46)
Table 3.1: Summary of material parameters indentified from monotonic loading/unloading tests
under five different strain rates.
.
Isochoric Part
of Network A
Rubber
AA
7.00MPa
Spring
JA
10.7
PA
4.42MPa
Damper
nA
0.0646
Linear
pB
.
82-3
Spring
Isochoric Part
of Network B
Viscous
Damper
Volumetric Part
3.4.2
QA
0.0447
nA
0.0755
K
829MPa
Model Calibration
The 1-D calibration process is composed of two steps: the seven parameters (i.e.
pA, JA,
PA, nA, pB, QB and nB) relating to isochoric deformation are identified assuming material
incompressibility (J = 1), before the bulk modulus r, is estimated based on the Lame moduli
and the Poison's ratio. A summary of all identified material parameters is given in Table 1.
Material Parameters of Network A (A,
JA, PA, nA)
The model is calibrated such that the response of Network A describes the rate dependency
of the hypothetical equilibrium path in the continuous compression experiments. As discussed
in Section 3.3.1 (see also Figure 3-6a), the hypothetical equilibrium path oA = OA (A, dA) for
a given strain rate amplitude dA is approximated by the average of the measured loading and
unloading stress-strain curves at a total strain rate
JA (A, JA)
2
e
[9A (A, JA) + OA (A, -
A)]
with
dA
W
The material parameters for Network A are calibrated in an iterative manner:
(3.47)
1. Assuming A' (t) ~ A (t), we estimate
(A,
JA)
from the best fit of Eq. (3.4344) to the
experimentally obtained stress-strain curve for the highest strain rate case (here, jaj
101 /s).
2. For the remaining strain rates, we calculate A' (t) based on Eq. (3.43) using (pA,
3. Using the relationship A' (t) = A (t) /A' (t), we calculate
JA).
a as a function of the strain-
like variable ( for each experiment (Figure 3-6b); in the figure, the curve labels indicate
the average effective viscous strain rate dA.
4. The parameters PA and nA are then obtained from the approximation Eq. (3.44) of the
curves shown in Figure 3-6c.
5. It is useful to evaluate the term
HA
HA :=
l0AI /
(exp (
-
1) from each experimental curve.
is a function of the effective viscous strain rate dA only and independent of the total
strain. In other words, each curve shown in Figure 3-6b reduces to a single data point
in Figure 3-6c. The subsequent curve fit
HA = (PA/J) (JA/dref)"A
of the data points in
Figure 3-6c yields the model parameters PA and nA.
After completing Steps 1 to 5, we obtain:
PA = 6.85MPa ,
JA = 11.0
PA = 4.44MPa
ni
,
(3.48)
= 0.0648
A second calibration loop is performed, starting with A' (t) = A (t) /A' (t) in Step 1, where
A' (t) is the viscous deformation predicted by the model parameters Eq. (3.48). Subsequent
evaluation yields:
7.OOMPa
,
JA = 10.7
PA = 4.42MPa
,
nA= 0.0646
PA =
(3.49)
The calibration procedure has been stopped at this point as additional iterations did not improve
the accuracy of the curve fit.
Approximate Network A Contribution
CU
6)
- -1
I-i
True Strain [-]
Network A: Calibration of P &n
Network A: Calibration of J7
6-
(b)
X~
42C
0.8
[-]
Test
101 10- 10- 10 10U 10 102 103 101
Eq. Viscous Deformation Rate, d [Isec]
Figure 3-6: Material model parameter identification for Network A. (a) Estimated experimental stress-strain curve for Network A; (b) Equivalent Mandel stress as a function of the scalar
deformation measure (; (c) HA as function of the equivalent viscous deformation rate dA for
the identification of the model parameters PA and nA through power-law fit.
Material Parameters of Network B (IB,
QB, nB)
The stress contribution of Network B is obtained by subtracting the identified stress contribution of Network A (using the parameters Eq. (3.49) in Eq. (3.44)) from the original
stress-strain curve for each strain rate
o-B (A,
where
= o) (A, )
-
o-A (A, jA)
(3.50)
dA corresponds to the average viscous strain rates that have been determined throughout
the calibration of Network A. Figure 3-7 shows the corresponding results for different strain
rates. According the proposed rheological model, viscoelastic behavior of Network B should
be symmetric with respect to the stress, i.e. O-B (A, -N) = -~B (A, a).
However, the curves
in Figure 3-7 illustrate that this symmetry assumption holds only approximately true when
using the calibrated analytical expression Eq. (3.44) for Network A. In close analogy with
the calibration procedure for Network A, we identify the model parameters associated with
Network B:
1
1. We assume Aes (t) ~ A (t) for the highest strain rate case (i.e. strain rate of 10 /s).
2. Determine pt
from the best curve fit for the experimental stress-strain curve for the
loading phase using Eq. (3.45).
3. Using pB, we can then calculate A'B (t) and Av (t) = A (t) /A' (t) for the other experiments.
The corresponding 0B Iversus ( curves are shown in Figure 3-7b. According to Eq. (3.46),
we have
o BI =V/-B
1 - exp
-(3.51)
where (B := QB (dB/dref)flB is a function of the strain rate only. Thus, we determine
(B from the approximation of Eq. (3.51) to the
IoI vs. ( curves.
4. Subsequently, the material constants QB and nB are found from plotting
of the effective viscous strain rate dB.
(B as a function
A second parameter identification loop is performed using A' (t) = A (t) /A' (t) leading to:
82.3MPa
ALB
QB = 0.0447
,
(3.52)
nA = 0.0755
Bulk Modulus of Material (r,)
The initial shear modulus for the proposed model is the sum of the shear modulus of each
network, i.e., P = P+ - pB = 89.3MPa. Experiments revealed that the Poisson's ratio of
v = 0.45 is nearly rate-independent throughout the deformation up to the strain of 1.0 (Shim
and Mohr, 2009a). Since the free energy for the volumetric deformation is based on the linear
elastic model, we can determine the bulk modulus using the relation between the elastic moduli
K =
3.5
2p (1 + v)
3 (1 --2v)
= 8.29MPa
(3.53)
Comparison of Simulation and Experiments
The constitutive model has been programmed as a user material subroutine for the finite element
software ABAQUS/Explicit. Using the identified material model parameters, we perform the
simulations of all experiments described in Section 3.2.
3.5.1
Continuous Compression
Figure 3-2c summarizes the numerically predicted stress-strain curves for different strain rates.
Similarly to the experiments (Figure 3-2b), the simulation results exhibit a high strain rate
sensitivity during the loading phases. In addition, the characteristic convergence of the stressstain curves for different loading velocities is also captured by the simulations. The comparisons
of experiment and simulation for individual strain rates are shown in Figures 3-8a to 3-8e. The
stress-strain curves show a better agreement during loading than during unloading. The loading
portions typically feature the largest error for small strains (less than 0.2) while good predictions
are achieved for large strains. The predicted response during unloading is too stiff for all strain
rates, i.e. the simulations reach a stress of zero at larger compressive strains than measured in
Approximate Network B Contribution
-0.8
-0.6 -0.4
True Strain [-]
Network B: Calibration of Q &n
Network B: Calibration of pB &gB
0.0
./
8
(b)~
1
0.0 6 -/
--
-d
d,~10'
-
Isec
- 101/aec
T
-,
-d
8
17/sec
d- 10-2/86C
0.0
16
4-
QI
0.02
0
C(!
0.4
0.8
4* I-I
1.2
1.6
0-""--n
""
Teat
"
10 10- 10- 10 10 10 102 10 10
Eqv. Viscous Deformation Rate, dB [Isec]
Figure 3-7: Material model parameter identification for Network B. (a) Estimated experimental stress-strain curve for Network B; (b) Equivalent Mandel stress as a function of the
scalar deformation measure (; (c) (B as function of the equivalent viscous deformation rate dB;
identification of the model parameters QB and nB.
the experiments. From an energetic point of view, it is worth noting that the differences during
the loading phase underestimate the mechanical work, while the work is overestimated during
unloading.
However, these two errors compensate each other when calculating the viscous
energy dissipation for the entire loading/unloading cycle. In other words, despite small errors
in the stress level during loading and unloading, the overall energy dissipation is well predicted
over a wide range of strain rates.
The graphs in Figure 3-8 also depict the predicted individual contributions of Network A
(dotted blue line) and Network B (dashed green line).
During the model calibration, we
assumed that the effective viscous strain rate dA is the same during loading and unloading,
which implies that Network A exhibits the same stress-strain response during loading and
unloading. However, since the total viscous strain A keeps on decreasing during unloading (the
driving stress for Network A is compression during both loading and unloading) even though
A increases, the decrease of the stress level is faster during unloading than the corresponding
increase during loading (hysteresis behavior of a Maxwell element). The differences between
the loading and unloading portion for Network A (blue dashed curves) illustrate this behavior.
The model's symmetric response of Network B (the green dashed curves show only the
loading portion) is characterized by a plateau over a wide range of strains. Since the elastic
strains in Network B are small, the driving stress of viscous deformation in Network B changes
rapidly from compression during loading to tension during unloading. As a result, the effective
rate of viscous deformation dB is the same during loading and unloading which is consistent with
the assumptions made during calibration. Thus, it is concluded that the differences between
the experiments and model predictions are associated with the response of Network A.
3.5.2
Stair Compression
Figure 3-3c provides an overview on the simulation results for all loading cases next to the
experimentally-measured stress-strain curves from stair compression tests. A direct comparison
of the experimental and the simulation results for each loading velocity is shown in Figure 39a to 3-9e. Overall, we observe the same model accuracy as for the monotonic loading and
unloading experiments, i.e. the model slightly underestimates the stress level for small strains
and predicts a faster decrease in stress level after reversal of the loading direction.
(b)
'
(a)
-5-
-5-
-10
-- 10
U)
U)
5-15
-
-15 --
-i%
-
-20-
.. S iulatio
-1
Test a
Simuion:
.a.
---
Sim ation: aA
--
Simulationa,
Simulation:
a.
-
-25
-
-
Test:a
Simulationoa +a,-
-
-20-
-0.8 -0.6 -0.4 -0.2
True Strain [-]
-25
0
-1
-0.8 -0.6 -0.4
True Strain
A+,
-0.2
0
-5-
-5 -
-1-10 U)
U)
U)
U)
-15
iZ-15 -
-
-25
(e)
-1
F-
Test:a
Simuaion aa
Simulation:
a
--
-20-
-
Test
20-
-
....... Simulation:
a
Simulation a
-0.8 -0.6 -0.4 -0.2
True Strain []
a
Simuation: ae,+a..
- - -Simulation:
a,
0
-25
-1
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
0
-5Lu
-10--15 H"
-20-
-Testa
--- Test: a
-
Simuion =a+a...... Simulationa
- - Simulation:
-25
-1
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
Figure 3-8: Comparison of simulation results and experiments for continuous loading-unloading
cycles. (a) 10- 3 /s, (b) 10 2 /s, (c) 10-1/s, (d) 100/s and (e) 101/s
(b)
-
(a)
J..
~-
-56-
-5 -
-- 10
-10-
-15 -
i-15 -
-20Simulation:
a
......... Simulation
a,
Snulation a.
-25
(c)
-1
(d)
.
-5-
- -
-25
0
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
-1
0
-
-15-
--
-20-
-
15-
Test: a
Simulationa=a +a, Simulationa
-20-
-0.8
-0.6
-0.4
True Strain []
0
-0.2
Test:a
Simuo
a-aA+a.
Simulationa
Simulation:
a
(e)
1
-0.2
-5-
-
-1
Simulation:
a,
-0.8 -0.6 -0.4
True Strain
0
-- 10 -i---10
-25
Test:a
imunion
-
Test:a
-20-
-25
- --
-1
-0.8
-0.6
, Simulation:a,
-0.2
-0.4
True Strain [-
0
-
0
-5--- 10r-15 H-Test:
a
Simulmon a- a +a,
-20-
. .Simulaion
a
Simulationa
-25
-1
o
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
Figure 3-9: Comparison of simulation results and experiments for stain loading/unloading. (a)
10-/s, (b) 10- 2 /s, (c) 10-1/s, (d) 100 /s and (e) 101 /s
The relaxation behavior of Network B is symmetric for loading and unloading (green dashed
lines in Figure 3-9).
Conversely, the stress-strain curves for Network A (blue dotted lines in
Figure 3-9) reveal that stress relaxation is more pronounced during the loading phase, while
unloading is mainly elastic. This non-symmetric feature of Network A is also visible at the
macroscopic level where the model predicts the experimental observation of a faster relaxation
during loading than during unloading. The model provides accurate estimates of the relaxationinduced stress decrease (jump) for small strains during the loading phase, while this decrease
is underestimated at large strains. This is attributed to the fact that the stress contribution
of Network B reaches its plateau level at small strains; as a result, the amplitude of stress
relaxation within Network B is very similar for small and large strains.
3.5.3
Relaxation
Figure 3-1b shows the predicted stress histories for all relaxation experiments.
In addition,
the histories of the relaxation modulus are presented in Figure 3-1d. The relaxation modulus
histories are well predicted for the entire range of strains. However, the comparison of the
true stress versus time curves indicates that the model predicts slower stress relaxation than
observed in the experiments. Similar conclusions may be drawn from the comparison of the
experimental and simulated isochronous stress-strain curves. Observe that the instantaneous
stress-strain curves agree well with the test results, while the long-term behavior (e.g. blue
curves for 616s) predicted by the model overestimates the stress level at large strains.
3.5.4
Discussion
The model is able to predict the strain rate dependent loading and relaxation behavior polyurea
with reasonable accuracy. Furthermore, the model provides good estimates of the rate dependent viscous dissipation throughout the loading/unloading cycles. In the present rheological
framework of two parallel Maxwell elements, the identification of the behavior of Network A
is important. Here, the response of Network A has been defined through the "rate-dependent
equilibrium path" concept; the mechanical response of Network A is identical throughout compressive loading and compressive unloading. Furthermore, in the particular case of polyurea,
this imaginary equilibrium path depends on the absolute value of the total true strain rate. The
response of the calibrated constitutive model illustrates that the assumption of a Maxwell model
does not lead to an exact description of the material behavior for both loading and unloading.
The hysteresis behavior for loading and unloading at a constant total strain rate is an inherent
feature of Maxwell models. Consequently, the rate dependency of the conceptual equilibrium
path (which is hysteresis free) cannot be represented accurately by a Maxwell model. An attempt has been made to calibrate the material model parameters using a numerical parameter
identification scheme (minimization of the error between the experiments and simulations). In
other words, the equilibrium path concept based separation of the contributions of Network A
and Network B has been omitted throughout the material model parameters identification.
However, the identified model response was not more satisfactory as improvements on the unloading response could only be obtained at the expense of the quality of the loading curve
prediction.
In order to come up with an improved constitutive model, one may consider a different rheological model for Network A, but to the best of the authors' knowledge, no thermodynamically
consistent model has been published in the open literature that can model the apparent elastic
(same loading and unloading curve), but yet strain rate dependent behavior of Network A.
Thus, alternative modeling approaches need to be explored in the future.
One thermody-
namically consistent option is the introduction of loading and unloading conditions for viscous
evolution. However, more research is needed at the microstructural (i.e. molecular) level to
justify such loading and unloading conditions from a mechanism point of view.
3.6
Conclusions
The large strain compression response of polyurea is investigated for strain rates ranging 10- 3 /s
from to 101/s. Continuous and stair-like loading and unloading experiments are performed in
addition to simple relaxation tests. The experimental results reveal that the so-called equilibrium path concept breaks down in the case of polyurea. In contradiction with the definition of
the equilibrium path, the identified average stress-strain curves in continuous loading/unloading
experiments depend on the rate of loading. Thus, as an alternative to the equilibrium path
based rheological models of the Zener type, a new constitutive model is formulated assuming
two parallel Maxwell elements. The finite strain constitutive equations are outlined in detail.
Subsequently, the eight material model constants are calibrated to describe the mechanical behavior of polyurea. The model predictions are in good agreement with the experimental results.
The characteristic convergence of the unloading paths for different strain rates is successfully
captured by the model. Furthermore, the rate dependent viscous dissipation throughout a loading/unloading cycle is described with reasonable accuracy. The predicted unloading behavior
is too stiff which is associated with the limitations of the Maxwell model that describes the soft
domain of polyurea.
Chapter 4
Validation Application
4.1
Introduction
Polyurea is a highly viscoelastic rubber material that is used for the impact protection of vehicle
structures. It is considered for the armor protection and retrofitting of military vehicles that are
exposed to the blast loading of improvised explosive devices. The anticipated effect of polyurea
coatings on the blast resistance of steel plates is twofold. Firstly, the polyurea can directly
absorb a portion of the blast energy as it undergoes large deformations. Secondly, the onset of
ductile fracture of a steel plate may be retarded through the use of a polyurea coating, thereby
increasing the energy absorption of the steel structure.
As discussed by Xue and Hutchinson (2008), necking occurs under uniaxial tension when
the average true stress becomes equal to the overall tangent hardening modulus (Considere
criterion). In the case of a coated ductile substrate, a high strain hardening coating material
can increase the effective hardening modulus of the bilayer material such that necking is retarded with respect to the Considere strain of the uncoated material. The bifurcation analysis
of Guduru et al. (2006) reveals that an added surface layer can increase the resistance of a
structural element to fragmentation. Moreover, their results show that the addition of a soft
coating with high strain hardening can improve the weight specific energy absorption of the
structural element. Xue and Hutchinson (2008) demonstrate that the ratio of the elastomer
modulus to the flow strength of the substrate controls the effect of necking retardation. McShane et al. (2008) performed tension and bulge tests on copper/polyurethane bilayers under
static and dynamic conditions. Their experimental measurements indicate that coatings do not
provide dynamic performance benefits on an equal mass basis. While the total blast resistance
increases, the weight specific energy absorption of the structure may actually decrease through
the application of a polymer coating. Dynamic ring expansion experiments have been performed
by Zhang et al. (2009) on polyurea coated aluminum 6061 - 0 and copper 101 at very high
strain rates (4000 - 15000/s). Their experimental results show that there is no significant effect
of the polyurea coating on the strain at the onset of localization.
It appears that the neck retardation effect in coated ductile substrates is difficult to achieve
when using polyurea in combination with typical engineering materials. However, as pointed
out by McShane et al. (2008), polyurea coatings may still be seen as a practical solution for
enhancing the blast resistance of metallic structures because of the ease of applying polyurea
on existing structures (retrofitting). Even though the performance of the steel substrate may
remain unaffected, a very thick polyurea layer can still increase the energy absorption in absolute
terms. The impulsive loading experiments of Amini et al. (2009a) reveal that polyurea coatings
have a strong effect on the energy transfer to the steel plate. In particular, they demonstrate
that the positioning of polyurea on the impact side promotes failure of the steel plate under
shock loading while a polyurea layer on the back of the plate attenuates the shock. In the present
paper, we deal with the prediction of the large deformation behavior of polyurea in structural
applications. Xue and Hutchinson (2008) made use of a Mooney-Rivlin model for the polymer
coating in their numerical analysis of the polymer/metal bilayers. Zhang et al. (2009) modeled
the behavior of polyurea using a nonlinear hyperelastic material model. However, both uniaxial
compression and tension tests have demonstrated that the mechanical response of polyurea is
highly strain-rate dependent (e.g. Amirkhizi et al., 2006, Roland et al., 2007, Sarva et al., 2007,
Shim and Mohr, 2009a). Amini et al. (2009b) make use of the temperature-, rate- and pressuresensitive constitutive model by Amirkhizi et al. (2006) to provide supporting simulation results
of their direct pressure pulse experiments.
Finite viscoelasticity models of elastomers may be formulated using the so-called hereditary
integral approach (Coleman and Noll, 1961, Bernstein et al., 1963, Lianis, 1963, McGuirt and
Lianis, 1970, Leonov, 1976, Johnson et al., 1994, Haupt and Lion, 2002, Amirkhizi et al., 2006)
but their validity is often limited to a narrow range of strain rates (Yang et al., 2000, Shim et al.,
2004, Hoo Fatt and Ouyang, 2007). As an alternative to the hereditary integral approach, the
framework of multiplicative decomposition of the deformation gradient (Kroner, 1960 and Lee,
1969) is frequently used in finite viscoelasticity (e.g. Sidoroff, 1974, Lubliner, 1985, Le Tallec
et al., 1993, Reese and Govindjee, 1998, Huber and Tsakmakis, 2000).
In that framework,
the nonlinear viscoelasticity of elastomers is commonly described through a rheological springdashpot models of the Zener type (e.g. Roland, 1989, Johnson et al., 1995, Bergstram and
Boyce, 1998, Huber and Tsakmakis, 2000, Quintavalla and Johnson, 2004, Bergstrom and
Hilbert, 2005, Qi and Boyce, 2005, Areias and Matous, 2008, Hoo Fatt and Ouyang, 2008,
Tomita et al., 2008).
In the present work, we present the time integration scheme for a newly developed ratedependent constitutive model for polyurea (Shim and Mohr, 2009b). After implementing the
model as a user material subroutine into a commercial finite element software, the model is used
to predict the mechanical response of thick polyurea layers under punch loading. Experiments
are performed on 10mm thick polyurea layers for different punch velocities and different hemispherical punch radii. It is found that the model provides an accurate description of the loading
phase, which validates the assumptions made with respect to strain-rate and pressure sensitivity. However, the predicted response deviates from the experimental result during unloading
which is discussed in detail.
4.2
4.2.1
Punch Experiments
Specimens
The polyurea specimens used in the study are extracted from a steel armor plate with a 12.7mm
thick layer of polyurea DragonShield-HT Explosive Resistant Coating (ERC). Rectangular samples of 46 x 40mm are cut from the coated armor plate using conventional machining. The coated
polyurea is not separated from the steel as the steel serves as specimen support throughout the
punching experiments. However, to guarantee a uniform layer thickness for all specimens, the
outer surface of the polyurea is machined down to a final polyurea thickness of 10mm. All
experiments are performed on polyurea in its virgin state (no prior loading) after a shelf life of
about four years.
4.2.2
Experimental Procedure
The specimens are clamped on the table of a hydraulic testing machine (Model 8080, Instron).
The specimens are loaded through hemispherical indenters that are attached to the moving
actuator of the upper crosshead. Two hemispherical indenters of different size are employed:
D = 12.7mm and D = 44.45mm (see Figures 4-ia and 4-1b). In addition, we consider two different types of lateral boundary conditions: (1) free in all lateral directions, and (2) constrained
in the width direction (Figures 4-1c and 4-1d). For the latter case, the polyurea specimen is
placed between two steel blocks which prevents bulging in the width direction, but does not
prevent possible shrinking of the specimen in width direction. The friction at the interface
between the indenters and the polyurea is reduced by grease and multiple 0.imm thick Teflon
layers (which are partially torn apart during the test). The experiments are performed under
displacement control at constant deceleration using the control software MAX (Instron, Canton). Starting with an initial velocity vo, the velocity-time profile decreases linearly until the
experiment is stopped at a velocity of -vo
(Figure 4-2). The initial position of the actuator
is chosen such that the loading direction is reversed (point vo
reaches 7mm. Experiments are performed for vo
=
0) when the punch depth
1mm/s and vo = 100mm/s. Throughout
all experiments, the punch force is measured using a 50kN load cell. The punch displacement
is measured using an LVDT that is integrated in the actuator. The overall stiffness of the testing frame of about I00kN/mm has obtained from the comparison of the LVDT readout with
optical displacement measurements (Shim and Mohr, 2009a). All displacements reported in the
following have been corrected by the deformation associated with the finite machine stiffness.
4.2.3
Experimental Results
Figure 4-3a summarizes the results for the experiments with the small hemispherical indenter
(D = 12.7mm). The measured force-displacement curves are monotonically increasing up to
the point of load reversal. During unloading, the force reaches zero at a displacement of about
3.2mm. Beyond that point, the punch moves faster than the surface of the creeping polyurea
specimen. The force level is about 40% higher in a punch test at vo = 100mm/s than that
at vo = 1mm/s which is mostly due to the rate dependent material behavior. Note that the
(a)
(b)
(c)
(d)
Figure 4-1: Photos of the experiments with (a) the small punch and (b) the large punch.
The second row shows the set up for (c) free boundaries in all lateral directions and (d) for
constrained boundary conditions in the width direction.
(a)
0.5
1
1.5
Normalized Time, tIvo/(2uo)| [-]
1.5
1
0.5
Normalized Time, tlvo/(2uo) H
Figure 4-2: Applied loading profiles: (a) Applied velocity history; the velocity axis is normalized
by the applied initial velocity of either vo = 1mm/s or 100m/s; the time axis is normalized by
12uo/vol with uo = 7mm; (b) Corresponding applied displacement history.
(a)
5 -Red:
OU
(b)
$,Mvul.Ion W Free B'C
-Tests
W ConsICrIned
BC.
- SmaiTests W Free BBC
25
v10mmls
Red:
Blue:
Blue: vo=1mm's
34-
-'
13-
-
100mmfs
1mm/s
7
0
0
20-
2 OI.
4.
0 1;0
0
6L
4
6
2
Displacement [mm]
5-
8
0
0
,
2
4
6
Displacement [mm]
8
Figure 4-3: Measured load-displacement curves for experiments with (a) the small punch, (b)
the large punch.
loading conditions are of quasi-static nature since the speed of elastic waves is polyurea (of
the order of 106 mm/s) is much faster than the loading velocities. It is interesting to observe
the convergence of the force-displacement curves for both velocities upon unloading which is
consistent with the results from uniaxial compression experiments (Shim and Mohr, 2009b).
The results with and without constraint in the width direction are almost identical.
Thus,
Figure 4-3a presents the results for free lateral boundary conditions only.
In close analogy with the results from the small punch, the measured force-displacement
curves for the large hemispherical punch experiments are loading velocity sensitive and show
a higher force level for the high loading velocity (Figure 4-3b).
Moreover, the large punch
experiments also show the characteristic convergence of the force-displacement curves during
unloading. For the large punch, the force level is zero at a punch displacement of 3.4mm. The
effect of the boundary condition in width direction becomes apparent when using the large
hemispherical punch (D = 44.45mm); the force level with the constraint (solid lines) is higher
than that with the free lateral boundary conditions (dotted lines). Regardless of the applied
velocity profiles, the constraint in the width direction increases the force level by about 3kN
at the maximum punching depth. Subtracting the force level with vo = 1mm/s from the force
level from vo
=
100mm/s for the corresponding displacement magnitude, one can find that the
NOn1rear
LiWar
Viscou
Elow. SpAng
Damper
Network B
Network A
Nordimar
Nonlin.ear
VimcousDamper
Rubber Spring
Figure 4-4: Rheological model of the rate dependent constitutive model for polyurea.
effect strain rate during loading phase results in the increase of force level up to 7kN about at
the punching depth of 7mm.
4.3
Constitutive Model
In the following, we present the algorithmic version of a recently-developed constitutive model
for polyurea. The reader is referred to Shim and Mohr (2009b) for details on the differential formulation and the underlying physical arguments for specific constitutive equations.
The algorithm is implemented as a user material subroutine for the finite element software
Abaqus/explicit. The constitutive equations are based on a rheological model of two Maxwell
elements that act in parallel (Figure 4-4). The first Maxwell element represents the soft part
(Network A) of polyurea and is composed of a nonlinear viscous damper and a nonlinear Gent
spring. The hard part (Network B) is represented by another nonlinear viscous damper and
a Hencky spring. The equations are cast in a framework of finite strains with multiplicative
Kroner-Lee decomposition (Kroner, 1960, and Lee, 1969) of the deformation gradient for each
Maxwell element. The specific evolution laws for individual model components are given in
algorithmic form below. The Euler forward numerical integration method is employed within
the subroutine for the effectiveness of the implementation, because of numerical challenges associated with Euler backward integration schemes for Maxwell model with nonlinear rubber
spring and nonlinear dashpot (e.g. Areias and Matous, 2008).
The material model comprises the viscous deformation gradients of Networks A and B, FA
and F',
as internal state variables. We consider the strain driven time integration problem,
where the variables at time
T
= t
+ At are calculated based on the solution at time t. In other
words, given the total deformation gradient Ftot (T) and the internal state variables F' (t) and
F' (t), we evaluate the total Cauchy stress T
(T)
along with the updated state variables F'
(T)
and F' (T). The hydrostatic part of the Cauchy stress tensor is directly related to the change
in total volume,
J (r) = det Ftot (r)
(4.1)
trT (T)
In J (T)
(4.2)
3
J (-r)
where K denoting the bulk modulus. The deviatoric part of the macroscopic stress tensor corresponds to the sum of the deviatoric stresses TA and TB acting on Network A and Network B,
respectively,
T (T)
trT (r)
=
devT (r) +
=
TA (T) + TB (T)
-
3
_
(4.3)
1
trT (r)
1
For the evaluation of the stresses and internal variables in Networks A and B, it is useful to
define the isochoric deformation gradient
F
(T) = [J (T)]-
1 3
/
(4.4)
Ftot (-r)
as well as the strain-like deformation measure
( (T) =
4.3.1
tr {F T (T) F (T)}
-
3
(4.5)
Response of Network A
The viscous gradient is calculated from the Euler forward form
Fv
(T)
= [1 + At Dv (t)] Fv (t)
(4.6)
where D' (t) is the rate of viscous deformation at time t. Subsequently, we have the isochoric
deformation gradient of elastic deformation in Network A
(r)
Using F'
(T),
(4.7)
= FA (t [F' (t)]-1
we determine the deviatoric Cauchy stress for Network A based on Gent's (1996)
free energy function
TA (
ytr
= "A)) 1 -(
{FeT (r) FeA (r)}
JA
J (r)
3
)-1
dev {Fe (r) F
T
(4.8)
(T)}
with the material parameters yA > 0 (initial modulus) and JA > 0 (locking stretch).
The viscous rate of deformation tensor D'
(T)
is obtained from the nonlinear viscous evo-
lution law. For this, we calculate the driving Mandel stress
MA (T) = J (r) dev
{F
(4.9)
T (T) TA (r) F e~T (T)}
along with the corresponding equivalent stress
nA (T) =
2
(4.10)
MA (T) : MA (T)
The equivalent rate of viscous deformation dA (T) is then given by the power-law
d~T~-
dA (r)
=_ dref
7'
A (1-)
M 0
PA [exp ((T) - 1]
i/nA
/ 1/A(4-11)
with the reference rate of deformation dref = 1/s and the material properties PA > 0 (viscosity
constant) and the exponent nA > 0 (constant for rate sensitivity). The flow rule assumes that
the rate of viscous deformation tensor D' is aligned with the driving Mandel stress MA
DA
3 dA r)
= 2r) -
MA (T)
4.3.2
Response of Network B
In close analogy with the procedure for Network A, the constitutive equations for Network B
are solved numerically. We have
Fv (w) = [1 + At Dv (t)] Fv (t)
(4.12)
Fe (r) = FB (t) [Fv (t)]- 1
(4.13)
and
The stiff elastic response of Network B is described by Hencky's strain energy function. Thus,
elastic right stretch tensor U'B (r) and the rotation tensor Re (r) are calculated from the polar
decomposition of the elastic deformation gradient,
F'
(T) = R'
(T) U'
with R T (r) R'
(T)
(T)
(4.14)
1
before calculating the deviatoric Cauchy stress
TB (T) = 2B R'
(r) [ln U'
(T)]
(4.15)
RT (r)
with the shear modulus pBSubsequently, we write
MB (T) = J (r) dev {F4T (r) TB (r) F'-
mnB (r)
MB
-
(T) :
MB
T
(4.16)
(r)}
(4.17)
(T)
The rate of viscous deformation is approximated by
-ndf( 1
(r
d
mB (T)
pV/'BQB
1
[p
(T)
QB
1/nB
(B___
dre
where QB and nB are material model parameters that control the rate sensitivity of Network B.
As for Network A, the rate of viscous deformation tensor D' is aligned with the driving Mandel
Table 4.1: Summary of material parameters indentified from monotonic loading/unloading tests
under five different strain rates.
.
Isochoric Part
of Network A
Isochoric Part
of Network B
7.OOMPa
Rubber
A
Spring
JA
10.7
.
Viscous
PA
4.42MPa
Damper
nA
0.0646
Linear
Spring
YB
82.3MPa
QA
0.0447
nA
0.0755
K
829MPa
Viscous
Damper
Volumetric Part
stress MB
DB
4.3.3
(T)
3 dB (r)
-
2
_
mnB
(T)
(4.19)
MB (T)
Model Parameter Identification
The constitutive model requires the identification of eight material parameters: four parameters
(pA,
JA, PA,
nA)
for Network A, three parameters
( pB, QA,
nA)
for Network B, and one
parameter (rz) describing the elastic volumetric response. All model parameters have been identified based on the results from uniaxial compression experiments at five different strain rates
between 10-3/s and 101 /s up to a true strain of -1.0.
Details on the parameter identification
procedure are given in Shim and Mohr (2009b). Table 4.1 summarizes the identified material
parameters which are used in the present structural validation study. The comparison of the
measured and predicted stress-strain curves for uniaxial compression is shown in Figure 4-5. It
is noted that the present choice of material model parameters provides a good description of the
initial monotonic loading response of polyurea, while the model predictions are systematically
too stiff during unloading.
(a)
(b)
U
-5-
-5-
-10 --
~-15
10
N-15--
-
-20-
....
Test:a
Testa
-20-
Simulaton a,+aaSimulation:a
-Sima+ono
..
- - - Simulation:
a
-25
(c)
-1
(d)
0
Simulation:
a,
-25
0
-0.8 -0.6 -0.4 -0.2
True Strain [-
-1
-0.2
0
-5-
-- 10
-10
U)
U)
-15 -
-
5-15
-
Test:a
-20-
-
a
-
-20-
-Simulation
--
Simulation:
a-
-1
-Test: a
-
a
Simulation a= aA+
Simulation.
(e)
-0.8 -0.6 -0.4
True Strain []
C
-5 -
-25
Simulation:
a,
1
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
-25
a-
. Simulation:
a
Simulation:
a,
-1
-0.8 -0.6 -0.4
True Strain
-0.2
0
0
-- 10U)
-15 STest:
a
-20-25
-
-1
1
Simulationa=a,+a,
- Simulation:aA
-Simulation:a,
-0.8 -0.6 -0.4 -0.2
True Strain [-]
0
Figure 4-5: Comparison of simulation results and experiments for continuous loading-unloading
cycles. (a) 10- 3 /s, (b) 10 2 /s, (c) 10- 1/s, (d) 100 /s, (e) 101/s.
4.4
Numerical Simulations of the Punch Experiments
Finite element simulations are performed of all punch experiments. We make use of the symmetry of the mechanical system by using a quarter model. The polyurea block is meshed with
eight-node reduced integration solid elements (type C3D8R of the Abaqus element library)
while using the user material option to describe the constitutive behavior. The punches are
modeled using rigid elements. The meshes comprise eight elements in thickness direction and
a small bias in the horizontal plane providing a smaller element size near the center than at
the specimen boundaries. Frictionless interface conditions are assumed between the punch and
the polyurea. The motion of all nodes at the bottom of the polyurea (which corresponds to the
interface with the steel substrate in the experiment) is set to zero. We applied the same punch
velocity histories as in the experiments and omitted addition of bulk viscosity.
The deformed meshes at the point of maximum penetration (no = 7mm) are shown in Figure
4-6 for the simulation without lateral constraint along the width direction. The comparison of
Figures 4-6a and 4-6b clearly shows that the bulging effect, i.e. the lateral expansion of the
polyurea block is more pronounced for the large than for the small punch. In the latter case,
we observe a small lateral displacement of about 0.5mm. The bulging has little effect in the
load-displacement curves for the small punch and we found nearly the same force-displacement
curve for free and constrained lateral boundary conditions.
The predicted force-displacement curves for the small punch are shown in Figure 4-7a.
The simulations results are in good agreement with the measured force-displacement curves
during the loading phase. A similar conclusion may be drawn from comparing the simulations
and experiments for the large punch (Figure 4-7b and 4-7c). The comparison shows that the
numerical model can predict (a) the effect of the punch size, (b) the effect of the lateral boundary
constraint, and (c) the effect of loading velocity on the force-displacement curve during loading.
Figure 4-8 shows the histories of the strain-like variable, (, and the viscous rates, dA and
dB, for the simulation of a small punch with vo = 100mm/s (the corresponding locations are
highlighted in Figure 4-6a). The profiles elucidate the strong variations in local strain rate
during the punch experiments. It is interesting to see that the local viscous rate can be as high
as 30/s which is close to the strain rate of the fastest calibration experiment (Figure 4-5e).
However, unlike for the calibration experiments, the strain rates are non-constant throughout
(a)
44.424
-01
2
-37730
(b)
+
c
2
Figure 4-6: The contour plots of the logarithmic strain in thickness-direction from simulations
with vo = 100m/s at an indentation depth of 7mm: (a) small hemispherical indenter, and (b)
large hemispherical indenter. The detail in Figure 4-6a shows the locations for which the strain
rates dA and dB are plotted in Figure 4-8.
a punch tests which is seen as an important validation of the model assumptions with respect
to the effect of strain rate. At a punching depth of 7mm, the simulation results overestimate
the force level by about 0.5kN in the case of the small punch and 2.5kN
in the case of the
large punch. This overestimation (which corresponds to less than 10% of the current force
level) is consistent with the model calibration results. Recall from Figure 4-5 that the model
systematically overestimates the stress level at very large compressive strains.
The model predictions during unloading deviate from the experimentally-measured forcedisplacement responses. The simulations predict the characteristic convergence of the loaddisplacement curves, but the predicted response during unloading is too stiff for both velocities
and both punch sizes.
Consequently, the instantaneous residual displacement at zero force
is overestimated in the simulations.
This deviation is again consistent with the calibration
experiments, indicating an inherent shortcoming of the model formulation for unloading.
4.5
Discussion on Unloading Behavior of Polyurea
The loading response of the model agrees well with the test results except for an overestimation
of the force at large punch displacements. The simulation results indicate that the material is
subject to strains of up to -2.0
as the punching depth reaches 7mm (see Figure 4-6). Since
this is twice as high as in the experiments for the material model parameter identification, we
performed additional material tests at a strain rate of 10-/s. Figure 4-9a shows the resulting
stress-strain curve from three different experiments. Each virgin-state specimen is subject to a
single loading-unloading cycle up to a maximum strain of -0.5, -1.0 and -1.5, respectively. The
corresponding simulations (Figure 4-9b) reveal that the numerically-predicted locking behavior
is more severe than that observed in the experiments.
The unloading path in the stress-strain curves comprises two characteristic regimes: a stiff
part at the beginning of unloading followed by a soft part as the stress approaches zero. Both
the experiments and the numerical simulations exhibit this feature. The numerical model does
not show the smooth transition between these two regimes, but this is seen as an acceptable
engineering approximation of the physical behavior. However, the apparent main deficiency of
the current model is its inability to capture the increase of the hysteresis loop width as the strain
Small Punch with Free B.C.
Displacement [mm]
Large Punch with Constrained B.C.
Large Punch with Free B.C.
6
4
2
Displacement [mm]
8
~0
6
4
2
Displacement [mm]
Figure 4-7: Comparison of simulations and experiments. Force-displacement curves for (a)
small punch with free lateral boundaries, (b) large punch with the free lateral boundaries, (c)
large punch with constraint in the width direction.
Simulations with
Small Punch and vo=100mm/s
i
4
in
3
4W_ 2
E
0
Time, t [s]
Simulations with
Small Punch and v,=100mm/s
(b)
-- -
(C)
Simulations with
Small Punch and v,=100mm/s
30
Location A: Top
---
LocationA: Top
LocationB:Midd
- - - Location C: Boat
LocationB: Middle
---
Location C: Bottom
20
20
0
z 4610
0
1'N
-
z
0U
W)
4-0 10
N-
0.07
0.14
0.21
Time, t [s]
0.28
\
0.14
Time, t [s]
0.21
0.28
Figure 4-8: Results from the small punch simulation with vo = 100mm/s and free lateral
boundary conditions at three different locations (labeled by A, B and C in Figure 4-6(a):
Histories of (a) the strain-like deformation measure, (, (b) the viscous strain rate of Network A,
dA, (c) the viscous strain rate of Network B, dB. Note that the loading direction is reversed
at t = 0.14s.
Simulations at 10-2/sec
Tests at 10-2/sec
(b)
(a)
-1 0 (D;
()
-20
-0s
-30
ef
-1.5
"
-20
-1
-0.5
True Strain [-]
05
-0
0o
0
05
-30-1.5
0
-1
-0.5
True Strain [-]
Figure 4-9: Stress-strain curves for a single loading-unloading cycle at
from (a) experiments and (b) simulations.
e=
10-
2
/s as obtained
increases. As the applied maximum strain increases, the simulations predict almost constant
stress drops (marked by a', b' and c' in Figure 4-9b) while the experiments show a substantial
increase in the magnitude of the stress drops as a function of strain (marked by a, b and c in
Figure 4-9a).
Recall that Network A in the current constitutive model is mainly responsible for the
rubbery behavior while Network B describes the high initial stiffness and the time-dependent
hysteresis. The contributions of Networks A and B are in opposite direction as far as the
hysteresis width is concerned. To shed more light on this particular feature, Figure 4-10 shows
a direct comparison of the model response and the experiments. In addition, we plotted the
individual contributions of Network A and Network B to the stress-strain curve (dashed curves
in Figure 4-10). Note that the magnitude of the stress contribution of Network B depends on
the compressive strain only, while its sign changes from compression to tension upon unloading.
The contribution of Network A on the other hand is a compressive stress irrespective of the
loading direction. Thus, as the compressive strain increases, Network B makes the hysteresis
wider while the opposite holds true for Network A.
In order to replicate the experimental
observation of a hysteresis width increase as a function of the maximum compressive strain
(Figure 4-9a), the contribution of Network B should be dominant. Moreover, the contribution
of Network B would need to increase as the compressive strain increases. However, due to the
specific choice of the viscous evolution law for Network B, the stress contribution of Network B
is more or less constant once the compressive strain has exceeded -0.1 (see the stress plateau
in the response curves for Network B in Figure 4-10).
An attempt was made to change
the calibration of the response of Network B using the present modeling framework, but the
subsequent simulation results were no longer satisfactory for the phase of loading.
A constitutive model with a different rheological composition needs to be used to improve
the predictions for unloading.
A previous study (Shim and Mohr, 2009b) has shown that
models of the Zener-type (spring in parallel with a single Maxwell element) cannot describe
the large deformation behavior of polyurea over a wide range of strain rates.
The present
results indicate that the assumption of two Maxwell elements in parallel provides an accurate
description for monotonic loading only. It can describe the two characteristic stiff and soft
regimes during unloading over a wide range of strain rates, but the model predictions are only
in poor quantitative agreement during unloading.
Throughout our model development, we focused on a single loading-unloading cycle on
virgin-state specimens. However, even though the loading of the material in its virgin state
appears to be the most important with respect to the real-life applications of polyurea, the
material response to cyclic loading may be instructive as far as the choice of the rheological
model is concerned. Figure 4-11 shows the stress-strain curve for polyurea for five consecutive
loading-unloading cycles. After each loading-unloading cycle at a constant true strain rate of
10- 2/s, we let the specimen creep at zero stress for about 300s before applying the subsequent
loading-unloading cycle. The significant difference between the first and subsequent loading
cycles illustrates the Mullins effect for polyurea. The stress-strain curve for the first loading is
characterized by the high initial stiffness and the high peak stress marked as A. After the first
loading-unloading cycle, however, the shape of the loading path changes noticeably while nearly
the same unloading path is observed.
Additional loading-unloading cycles create very little
changes in both loading and unloading. This observation suggests that the constitutive model
should comprise an internal variable that reflects the amount of "microstructural damage"
associated with the Mullins effect. The Mullins effect has been investigated by many research
groups using either damage-based constitutive models (e.g., Simo, 1987, Govindjee and Simo,
6ef=-0.5
0
0U)
i -202-
-
Test a
Simulation: a- aA+a
---
-30
-
-
-
-1.5
Simutatin
a
Simulation a
0
-0.5
-1
True Strain [-]
6
-1.0
0
C-
0-
0)
- 20-
-3 0
Test: a
Simulationw:
a= a,+
a
Simution.aA
---
Simulaton: a,
-1.5
-0.5
-1
True Strain [-]
C
E6=-1.5
-10
Cn
-20
F-
-30
-0.5
-1
True Strain [-]
Figure 4-10: Comparison of simulation results and experiments for continuous loading/unloading cycles at a = 10- 2 /s. (a) ef = -0.5, (b) ef = -1.0, (c) ef = -1.5.
Tests at 102/sec
0
10F-
-
No. df OCcks-I
-No
of Ccls2
-No.
of Cycles=3
-No. of CycleS-4
-- NO.OfCycles_5~
-15 -
-1
-0.8
-0.6
-0.4
True Strain [-]
-0.2
0
Figure 4-11: Illustration of the Mullins effect. Five compression loading and unlading cycles
are performed at the constant strain rate of 10- 2 /s up to the maximum strain of -1.0. The
stress is zero between subsequent cycles for about five minutes.
1991, 1992, Lion, 1996, 1997, Miehe and Keck, 2000) or the concept of hard/soft domain
reorganization (e.g. Johnson and Beatty, 1993a, 1993b, Beatty and Krishnaswamy, 2000, Qi
and Boyce, 2004, 2005). Here, the explicit account of the Mullins effect is deferred to future
work since further experimental data is needed to analyze the effect of loading velocity over a
wide range of strain rates.
4.6
Conclusion
Punch experiments have been performed on 10mm thick polyurea samples at punch velocities
of up to 100mm/s which resulted in maximum local viscous strain rates of up to 30/s inside the
polyurea layers. A newly-developed rate-dependent constitutive model for polyurea has been
implemented into a finite element program and used to predict the experimentally-measured
force-displacement curves for different punch sizes and velocities. Furthermore, the effect of
lateral constraints has been investigated. The model provides accurate predictions of the loading response for all six test configurations which is interpreted as a partial validation of the
assumptions made with respect to strain-rate and pressure sensitivity. During unloading, the
model exhibits the characteristic stiff and soft responses of polyurea; however, it systematically
overestimates the depth of the quasi-instantaneous residual punch imprint after unloading.
Chapter 5
Conclusion and Suggestions
5.1
Summary of Main Results
The main results of this dissertation can be summarized twofold: experiments and constitutive
modeling. First, the effect of strain rate on polyurea are experimentally quantified for a wide
range of strain rates, i.e. 10-/s
~ 4000/s. The modified SHPB system of Zhao and Gary
(1997) has been used to perform compression tests on polyurea at low, intermediate and high
strain rates (10/s ~ 1000/s). It is composed of nylon input and output bars, while the striker
bar is substituted by a hydraulic actuator. Using the deconvolution technique by Bussac et al.
(2002), the time limitation of conventional SHPB systems may be overcome, thereby enabling
the use of the modified SHPB system for low and intermediate strain rate experiments of
long duration. The experiments confirm the known strain rate sensitivity of polyurea. The
measured stress levels correspond well to earlier results which have been obtained from tests
on conventional SHPB systems with very long bars.
Second, a new constitutive model is proposed to capture the rate-sensitivity for a wide range
of strain rates and the characteristic high stiffness behavior of virgin polyurea at small strains.
Various types of loading conditions are tested for the purpose of validation, and their results
are well compared with the model predictions in loading phases. The large strain compression
response of polyurea is investigated for strain rates ranging 10
3
/s from to 101/s. Continuous
and stair-like loading and unloading experiments are performed in addition to simple relaxation
tests. The experimental results reveal that the so-called equilibrium path concept breaks down
in the case of polyurea. In contradiction with the definition of the equilibrium path, the identified average stress-strain curves in continuous loading/unloading experiments depend on the
rate of loading. Thus, as an alternative to the equilibrium path based rheological models of the
Zener type, a new constitutive model is formulated assuming two parallel Maxwell elements.
The finite strain constitutive equations are outlined in detail, and it has been implemented
into a finite element program. Subsequently, the eight material model constants are calibrated
to describe the mechanical behavior of polyurea. The model predictions are in good agreement with the experimental results. The characteristic convergence of the unloading paths for
different strain rates is successfully captured by the model. Furthermore, the rate dependent
viscous dissipation throughout a loading/unloading cycle of uniaxial conditions is described
with reasonable accuracy. For the validation of the proposed model, punch experiments have
been performed on 10mm thick polyurea samples at punch velocities of up to 100mm/s which
resulted in maximum local viscous strain rates of up to 30/s inside the polyurea layers. The
proposed constitutive model is used to predict the experimentally-measured force-displacement
curves for different punch sizes and velocities. Furthermore, the effect of lateral constraints has
been investigated. The model provides accurate predictions of the loading response for all six
test configurations which is interpreted as a partial validation of the assumptions made with
respect to strain-rate and pressure sensitivity.
5.2
Suggestions for Future Studies
The following topics can be suggested for future studies:
" Constant strain rates from the hydraulic SHPB: Although the intrinsic time limitation
of SHPB systems could be overcome, this study also shows that it is still not possible
to perform experiments at reasonably constant strain rates with this technique. This is
due to the finite length (i.e. 3m in this study) of the input and output bars which causes
a periodic change in loading velocity. It is shown that intermediate strain rate SHPB
experiments require either very long bars (> 20m) or very short bars (< 0.5m) in order
to achieve an approximately constant strain rate throughout the entire experiment.
* Improvement in the fixed boundary conditions in the hydraulic SHPB: In order to prevent
the failure of the nylon bars under excessive loads, a fixed end support system in the
hydraulic SHPB of the current study is designed such that the bars are released before
elastic buckling occurs. However, with the current design of the fixed end, significant
force drops are found during tests due to the premature partial failure of the fixed end
support of the output bar. A new design of the fixed end support can be investigated to
prevent the elastic buckling of the bars and to resist the specimen strength at the same
time.
e Improvement in the unloading prediction of the constitutive model: From various tests
with different loading conditions, the model exhibits the characteristic stiff and soft responses of polyurea during unloading. However, the predicted unloading behavior is too
stiff, and the model systematically overestimates the depth of the quasi-instantaneous
residual strain/displacement after unloading.
A new rheological model can be investi-
gated to improve the prediction of unloading behavior and to capture the Mullins effect
for polyurea.
Appendix A
Identification of the Wave
Propagation Coefficient for
Viscoelastic Bars
The complex-valued propagation coefficient
(
(w) is a function of both the geometric and ma-
terial properties of the bars. If the complex modulus of the viscoelastic bar material is known,
( (w)
may be calculated from solving Pochhammer-Chree's frequency equation (Zhao and Gary,
1995). As an alternative, we make use of an experimental method that considers both geometric
dispersion and viscous attenuation within the framework of the 1-D wave theory (e.g. Bacon,
1998, Lundberg and Blanc, 1988) to determine
(
(w). Recall the solution of the one-dimensional
wave equation for viscoelastic bars,
s (x, W) =
where
((w)
sR (x, w)
and
sL
eR
(w) e*i(w)x + sL (w) ei'(wx
(A.1)
(x, w) are the rightward and the leftward traveling strain waves. Here,
is the propagation coefficient of the bars defined by
+ id (W) =
W + i(A.2)
c (w)
where
n (w) is the wave number, c (w) is the frequency-dependant longitudinal wave propagation
speed, and d (w) < 0 represents the attenuation coefficient. Note that c (w) and d (w) are even
functions of w while r, (w) is an odd function in the frequency space.
After performing an impact test on a single bar, we measure the incident and reflected
strain histories (i.e. einc (x = 0, t) and eef (x = 0, t)) at x = 0 within in the bar. In order to
determine the transfer function of the bar, we use the two facts from the impact test:
* The rightward and the leftward travelling strain waves are equal to the measured incident
and reflected strain histories, respectively:
sR (W) =
6
L
(w) =
sinc (x
ref
= 0, w)
(A.3)
(X = 0, w)
(A.4)
" The normal force is zero at the free end of the bar (i.e. the opposite end to the impacting
end of the bar):
sR (W) eif()d
+ sL (w)
eif(w)d = 0
(A.5)
where x = d is the distance between the strain gage location and the free end of the bar.
Now, the transfer function of the bar H (w) is experimentally determined from
H (
6
ref (x = 0,
w)
-
inc (x = 0, w)
e-
2
if()d
(A.6)
From Eq. (A.6), two components of the propagation coefficient can be identified using the
relations
ic Mc)
arg [H (w)]
2d
(w)
M~
ln {1H (w)]
2d
(A.7)
(A8)
Appendix B
List of Papers with Reference to
Respective Chapters
The following papers are published or submitted for publication with reference to the respective
chapters.
Chapter 2
Shim, J. and Mohr, M. (2009) "Using split Hopkinson pressure bars to perform large strain
compression tests on polyurea at low, intermediate and high strain rates," InternationalJournal
of Impact Engineering, 36:1116-1127.
Chapter 3
Shim, J. and Mohr, M. (2009) "Finite strain constitutive model of polyurea for a wide range
of strain rates," (submitted for publication, September 2009).
Chapter 4
Shim, J. and Mohr, M. (2009) "Punch indentation of polyurea at different loading velocities:
Experiments and numerical simulations," (submitted for publication, October 2009).
100
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